Chapter 9
CHAPTER 9: FACTORING EXPRESSIONS AND SOLVING BY FACTORING Chapter Objectives By the end of this chapter, students should be able to Factor a greatest common factor Factor by grouping including rearranging terms Factor by applying special-product formulas Factor trinomials by using a general strategy Solve equations and applications by factoring
SECTION 9.1: GREATEST COMMON FACTOR AND GROUPING ...... 242 A. FINDING THE GREATEST COMMON FACTOR ...... 242 B. FACTORING THE GREATEST COMMON FACTOR ...... 244 C. A BINOMIAL AS THE GREATEST COMMON FACTOR ...... 246 D. FACTOR BY GROUPING ...... 246 E. FACTOR BY GROUPING BY REARRANGING TERMS ...... 247 EXERCISE ...... 248 SECTION 9.2: FACTORING TRINOMIALS OF THE FORM x2 + bx + c ...... 249 A. FACTORING TRINOMIALS OF THE FORM x2 + bx + c ...... 249 B. FACTORING TRINOMIALS OF THE FORM x2 + bx + c WITH A GCF ...... 251 EXERCISE ...... 252 SECTION 9.3: FACTORING TRINOMIALS OF THE FORM ax2 + bx + c ...... 253 A. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c BY GROUPING ...... 253 B. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c BY THE “BOTTOMS UP” METHOD ...... 254 C. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c BY THE TRIAL AND ERROR METHOD ...... 255 D. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c WITH A GCF IN THE COEFFICIENTS ...... 256 EXERCISE ...... 257 SECTION 9.4: SPECIAL PRODUCTS ...... 258 A. DIFFERENCE OF TWO SQUARES ...... 258 B. PERFECT SQUARE TRINOMIALS ...... 259 C. FACTORING SPECIAL PRODUCTS WITH A GCF IN THE COEFFICIENTS ...... 260 D. A SUM OR DIFFERENCE OF TWO CUBES ...... 260 EXERCISE ...... 263 SECTION 9.5: FACTORING, A GENERAL STRATEGY...... 264 EXERCISE ...... 265 SECTION 9.6: SOLVE BY FACTORING ...... 266 A. ZERO PRODUCT RULE ...... 266 B. SOLVE EQUATIONS BY FACTORING ...... 267 C. SIMPLIFY THE EQUATION ...... 272 EXERCISE ...... 273 CHAPTER REVIEW ...... 274
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Chapter 9
SECTION 9.1: GREATEST COMMON FACTOR AND GROUPING A. FINDING THE GREATEST COMMON FACTOR In this lesson, we focus on factoring using the greatest common factor, GCF, of a polynomial. When we multiplied polynomials, we multiplied monomials by polynomials by distributing, such as
4 (2 3 + 8) = 8 12 + 32 2 2 4 3 We work out the same problem, 𝑥𝑥but backwards𝑥𝑥 − 𝑥𝑥 . We will𝑥𝑥 start− with𝑥𝑥 8 𝑥𝑥 12 + 32 and obtain its factored form. 2 3 𝑥𝑥 − 𝑥𝑥 First, we have to identify the GCF of a polynomial. We introduce the GCF of a polynomial by looking at an example in arithmetic. The method in which we obtained the GCF between numbers in arithmetic is the same method we use to obtain the GCF with polynomials.
Definition The factored form of a number or expression is the expression written as a product of factors.
The greatest common factor (GCF) of a polynomial is the largest polynomial that is a factor of all terms in the polynomial.
MEDIA LESSON Determine the GCF of Two Monomials (Duration 2:32)
View the video lesson, take notes and complete the problems below
Find the GCF of 88 and 24 . 18 13 𝑟𝑟 𝑟𝑟 88 24 /\ /\ 88 = ______
18
24𝑟𝑟 = ______
13
GCF𝑟𝑟 = ______
YOU TRY
Find the GCF a) 24 and 56 b) 12 , 6 and 21 3 15 5 20 7 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑦𝑦
242
Chapter 9 MEDIA LESSON Determine the GCF of two monomials (Two variables) (Duration 3:45)
View the video lesson, take notes and complete the problems below
Find the GCF of 108 and 96 . 5 3 7 2 108 96 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 /\ /\
108 = ______
5 3
96 𝑥𝑥 𝑦𝑦______
7 2
GCF=______𝑥𝑥 𝑦𝑦
YOU TRY
Find the GCF: a) 15 and 45 b) 24 , 18 , and 12 3 6 4 2 4 2 2 4 3 5 𝑚𝑚 𝑛𝑛 𝑚𝑚 𝑛𝑛 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦𝑧𝑧
MEDIA LESSON Find common factor with smaller coefficients (Duration 2:28)
View the video lesson, take notes and complete the problems below
Greatest common factor: ______
On variables we use______
Example: Find the greatest common factor. a) 15 + 10 25 b) 4 12 + 20 4 2 5 4 7 2 6 9 𝑎𝑎 𝑎𝑎 − 𝑎𝑎 𝑎𝑎 𝑏𝑏 − 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑏𝑏
YOU TRY
Find the GCF: a) 4 20 + 10 b) 6 15 + 9 2 4 3 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑥𝑥
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Chapter 9 B. FACTORING THE GREATEST COMMON FACTOR MEDIA LESSON Factor using the product method (Duration 6:39)
View the video lesson, take notes and complete the problems below
Identify the greatest common factor. Then factor. a) 18 + 6 + 24 185 4 3 6 24 𝑥𝑥/\ 𝑥𝑥 𝑥𝑥 /\ /\ GCF = ______
______
b) 60 15 + 45 604 3 3 4 2 5 15 48 𝑎𝑎/\𝑏𝑏 − 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 /\ /\ GCF = ______
______
______
YOU TRY
Factor: a) 25 15 + 20 b) 12 20 16 4 6 3 3 2 4 3 5 4 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑧𝑧 − 𝑥𝑥 𝑦𝑦 𝑧𝑧 − 𝑥𝑥𝑦𝑦
244
Chapter 9 MEDIA LESSON Factor using the division method (Duration 4:08)
View the video lesson, take notes and complete the problems below a (b + c) = ______
Put ______in the front, and divide. What is left goes in the ______.
Example: Factor a) 9 12 + 6 b) 21 14 + 7 4 3 2 4 5 3 7 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑎𝑎 𝑏𝑏 − 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏
YOU TRY Factor using the division method.
a) 21 + 14 + 7 b) 4 20 + 16 3 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
CHECK YOUR SOLUTION: To check your answer, you can distribute your GCF back into the parenthesis.
MEDIA LESSON Using distributive property to verify the factored form (Duration 2:46)
View the video lesson, take notes and complete the problems below
Example: a) 5 (6 2 + 5) b) 3 (6 + 2 7) 2 2 4 3 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 −
YOU TRY
Check your answer from the YOU TRY in the section Factoring using the division method above.
a) b)
245
Chapter 9 Steps for factoring out the greatest common factor Step 1. Find the GCF of the expression. Step 2. Rewrite each term as a product of the GCF and the remaining factors. Step 3. Rewrite as a product of the GCF and the remaining factors in parenthesis. Step 4. ✓Verify the factored form by multiplying. The product should be the original expression.
C. A BINOMIAL AS THE GREATEST COMMON FACTOR As part of a general strategy for factoring, we always look for a GCF. Sometimes the GCF is a monomial, like in the previous examples, or a binomial. Here we discuss factoring a polynomial where the GCF is a binomial. MEDIA LESSON Binomial GCF (Duration 2:20)
View the video lesson, take notes and complete the problems below
GCF can be a ______.
Example: a) 5 (2 7) + 6 (2 7) b) 3 (2 + 1) 7(2 + 1)
𝑥𝑥 𝑦𝑦 − 𝑦𝑦 𝑦𝑦 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
YOU TRY
Factor: a) 3 (2 + 5 ) 4 (2 + 5 ) b) (9 2)3 (9 2)5 2 2 𝑎𝑎 𝑎𝑎 𝑏𝑏 − 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑥𝑥 − 𝑦𝑦 − 𝑥𝑥 − 𝑥𝑥
D. FACTOR BY GROUPING When we have polynomials that have at least 4 terms. Sometimes, we can factor them by using a process known as factor by grouping.
Steps for factoring by grouping
To factor by grouping, we first notice the polynomial expression obtains four terms. Step 1. Group two sets of two terms, e.g., ax + ay + bx + by = (ax + ay) + (bx + by). Step 2. Factor the GCF from each group, e.g., a(x + y) + b(x + y) Step 3. Factor the GCF from the expression, e.g., (x + y)(a + b).
MEDIA LESSON Factoring by grouping (Duration 4:01)
View the video lesson, take notes and complete the problems below
Grouping: GCF of the ______and ______Then factor out ______(if it matches!) 246
Chapter 9 Example: a) 15 + 10 18 12 b) 6 + 3 + 2 + 2 𝑥𝑥𝑥𝑥 𝑦𝑦 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 𝑦𝑦
YOU TRY
Factor by grouping: a) 10 + 15 + 4 + 6 b) 6 + 9 14 21 2 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦
c) 5 8 10 + 16 d) 12 14 6 + 7
𝑥𝑥𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦 𝑎𝑎𝑎𝑎 − 𝑎𝑎 − 𝑏𝑏
E. FACTOR BY GROUPING BY REARRANGING TERMS Sometimes after completing Step 2, the binomials are not identical (by more than a negative sign). At this point, we must return to the original problem and rearrange the terms so that when we factor by grouping, we obtain identical binomials in Step 2.
MEDIA LESSON Factor by grouping – rearranging terms (Duration 4:41)
View the video lesson, take notes and complete the problems below
If binomials don’t match: ______
Example: a) 12 7 + 3 28 b) 6 20 + 8 15 2 𝑎𝑎 − 𝑏𝑏 𝑎𝑎𝑎𝑎 − 𝑎𝑎 𝑥𝑥𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦
YOU TRY
Factor: a) 4 21 + 6 14 b) 8 12 + 15 10 2 3 2 𝑎𝑎 − 𝑏𝑏 𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑏𝑏 𝑥𝑥𝑥𝑥 − 𝑦𝑦 − 𝑥𝑥
247
Chapter 9 EXERCISE
Factor the greatest common factor (GCF) if possible. If not, write “No common factor”. Check your answer by multiplying the factors.
1) 9 + 8 2) 45 25 2 2 𝑏𝑏 𝑏𝑏 𝑥𝑥 − 3) 56 35 4) 7 35 2 2 − 𝑝𝑝 𝑎𝑎𝑎𝑎 − 𝑎𝑎 𝑏𝑏 5) 3 + 6 6) 5 5 15 2 3 2 2 3 4 − 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 7) 20 30 + 30 8) 28 + 40 + 8 4 4 3 𝑥𝑥 − 𝑥𝑥 𝑚𝑚 𝑚𝑚 9) 30 + 5 15 10) 48 56 50 9 2 2 2 3 5 𝑏𝑏 𝑎𝑎𝑎𝑎 − 𝑎𝑎 − 𝑎𝑎 𝑏𝑏 − 𝑎𝑎 𝑏𝑏 − 𝑎𝑎 𝑏𝑏 11) 20 + 15 + 35 12) 3 + 5 6 8 2 2 5 5 3 3 8 − 𝑚𝑚𝑛𝑛 − 𝑚𝑚 13) 30𝑥𝑥 𝑦𝑦 𝑧𝑧 5 +𝑥𝑥5 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 14) 50 + 10 + 70 2 2 2 𝑞𝑞𝑞𝑞𝑞𝑞 − 𝑞𝑞𝑞𝑞 𝑞𝑞 𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥 𝑥𝑥𝑧𝑧 15) 1 + 2 16) 18 + 3 21 + 3 2 5 3 3 2 𝑛𝑛 − 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 − 𝑚𝑚 𝑛𝑛 𝑚𝑚𝑚𝑚
Factor completely. Check your answer using distributive property.
17) 40 8 25 + 5 18) 3 2 9 + 6 3 2 3 2 𝑟𝑟 − 𝑟𝑟 − 𝑟𝑟 𝑛𝑛 − 𝑛𝑛 − 𝑛𝑛 19) 15 + 21 35 49 20) 3 + 15 + 2 + 10 3 2 3 2 𝑏𝑏 𝑏𝑏 − 𝑏𝑏 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 21) 35 28 20 + 16 22) 7 49 + 5 35 3 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 23) 32 + 40 + 12 + 15 24) 16 56 + 2 7 2 𝑥𝑥𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 25) 2 8 + 7 28 26) 40 + 35 8 7 2 3 2 2 𝑥𝑥𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 𝑦𝑦 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 − 𝑦𝑦 − 𝑦𝑦 27) 35 10 56 + 10 28) 16 3 6 + 8 3 2 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 29) 6 48 + 5 40 30) 14 + 10 7 5 3 2 3 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑣𝑣 𝑣𝑣 − 𝑣𝑣 − 31) 7 + 21 5 15 32) 28 + 21 + 20 + 15 3 2 3 2 𝑛𝑛 𝑛𝑛 − 𝑛𝑛 − 𝑝𝑝 𝑝𝑝 𝑝𝑝 33) 15 2 6 + 5 34) 42 21 49 + 18 2 3 3 2 35) 5 𝑎𝑎𝑎𝑎 −10𝑏𝑏+−2 𝑎𝑎 25𝑏𝑏 36) 8𝑟𝑟 −+ 15 − 40𝑟𝑟 + 3 𝑟𝑟
𝑚𝑚𝑚𝑚 − 𝑚𝑚 − 𝑛𝑛 − 𝑚𝑚 𝑛𝑛 − 𝑚𝑚𝑚𝑚 37) 4 + 14 + 12 + 42 38) 56 + 8 7 2 𝑢𝑢𝑢𝑢 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑥𝑥 − 𝑦𝑦 𝑥𝑥𝑥𝑥 − 39) 56 + 14 49 16 40) 24 + 25 20 30 2 3 𝑎𝑎𝑎𝑎 − 𝑎𝑎 − 𝑏𝑏 𝑥𝑥𝑥𝑥 𝑦𝑦 − 𝑥𝑥 − 𝑦𝑦 248
Chapter 9 SECTION 9.2: FACTORING TRINOMIALS OF THE FORM x2 + bx + c A. FACTORING TRINOMIALS OF THE FORM x2 + bx + c Factoring with three terms, or trinomials, is the most important technique, especially in further algebra. Since factoring is a product of factors, we first look at multiplying to develop the process of factoring trinomials.
Steps for factoring trinomials of the form + + 𝟐𝟐 Step 1. Find two numbers, and 𝒙𝒙, that𝒃𝒃 𝒙𝒙 + 𝒄𝒄 = and =
Step 2. Rewrite the expression𝒑𝒑 so 𝒒𝒒that the𝒑𝒑 middle𝒒𝒒 term𝒃𝒃 is𝒑𝒑 split⋅ 𝒒𝒒 into𝒄𝒄 two terms, and . Step 3. Factor by grouping. 𝒑𝒑 𝒒𝒒
Step 4. ✓ Verify the factored form by finding the product.
MEDIA LESSON Factoring a trinomial with leading coefficient of 1 (ac method) (Duration 10:33 )
View the video lesson, take notes and complete the problems below
Factoring trinomials with a leading coefficient of 1. + + 1. Make two sets of parentheses and put the𝟐𝟐 factors of in the first position of each set of parentheses. 𝒙𝒙 𝒃𝒃𝒃𝒃 𝒄𝒄 2 (x )(x 𝑥𝑥 )
2. The second positions are the factors of c that add to b.
Example: Factor. Factors of c Factors of c a) + 8 + 12 ______b) 4 32 2 2 ______𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −
c) 13 + 36 d) + 12 2 2 𝑦𝑦 − 𝑦𝑦 𝑥𝑥 𝑥𝑥 −
e) + 28 + 75 f) 3 27 + 42 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
g) 2 12 80 3 2 𝑦𝑦 − 𝑦𝑦 − 𝑦𝑦
249
Chapter 9 MEDIA LESSON Factoring trinomials 2 (Duration 5:01 )
View the video lesson, take notes and complete the problems below
Factor: a) + 12 + 32 b) + 11 60 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 −
c) 9 + 20 d) 5 24 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −
YOU TRY
a) + 9 + 18 b) 4 + 3 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
c) 8 20 d) 9 + 14 2 2 2 𝑥𝑥 − 𝑥𝑥 − 𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑏𝑏
MEDIA LESSON Factoring trinomials – “X box” method (Duration 3:21)
View the video lesson, take notes and complete the problems below
If there is a ______in front of , then the ac method gives us ______. 2 a) 2 8 𝑥𝑥 b) + 7 8 2 2 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑦𝑦
250
Chapter 9 YOU TRY
a) 8 + 15 b) + 2 8 2 2 2 2 𝑢𝑢 − 𝑢𝑢𝑢𝑢 𝑣𝑣 𝑚𝑚 𝑚𝑚𝑚𝑚 − 𝑛𝑛
PRIME POLYNOMIALS: If a trinomial (or polynomial) is not factorable, then we say we the trinomial is prime.
For example: factor + 2 + 6. 2 We identify b = 2 and𝑥𝑥 c = 6 𝑥𝑥 Factor of c Sum 2, 3 2 + 3 = 5, not b −2, −3 −2 + −3 = −5, not b 1, 6 1 + 6 = 7, not b −1, −6 −1+ −6 = −7, not b
We can see from the table that there are not any factors of 6 whose sum is 2. In this case, we call this trinomial not factorable, or better yet, the trinomial is prime.
B. FACTORING TRINOMIALS OF THE FORM x2 + bx + c WITH A GCF Factoring the GCF is always the first step in factoring expressions. If all terms have a common factor, we first, factor the GCF and then factor as usual.
MEDIA LESSON Factoring trinomials with factoring GCF first (Duration 3:39)
View the video lesson, take notes and complete the problems below a) 7 + 21 70 b) 4 + 36 + 80 2 4 3 2 2 3 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦
YOU TRY
Factor: a) 3 24 + 45 b) 4 + 52 + 168 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥
251
Chapter 9 EXERCISE Factor completely. If a trinomial is not factorable, write “prime”.
1) + 17 + 72 2) 9 + 8 2 2 𝑝𝑝 𝑝𝑝 𝑛𝑛 − 𝑛𝑛 3) 9 10 4) + 12 + 32 2 2 𝑥𝑥 − 𝑥𝑥 − 𝑏𝑏 𝑏𝑏 5) 9 10 6) 15 + 56 2 2 𝑥𝑥 − 𝑥𝑥 − 𝑛𝑛 − 𝑛𝑛 7) + 15 + 54 8) 8 + 15 2 2 𝑝𝑝 𝑝𝑝 𝑛𝑛 − 𝑛𝑛 9) 11 + 18 10) + 12 2 2 2 2 𝑥𝑥 − 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑦𝑦 11) + 4 12 12) 5 + 60 + 100 2 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 − 𝑦𝑦 𝑎𝑎 𝑎𝑎 13) 6 + 24 192 14) 6 + 18 + 12 2 2 2 𝑎𝑎 𝑎𝑎 − 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 15) 6 + 96 + 378 16) 72 2 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑥𝑥 − 𝑥𝑥 − 17) + 30 18) + 13 + 40 2 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 19) + 13 36 20) 6 36 162 2 2 2 𝑚𝑚 𝑥𝑥 − 𝑚𝑚 − 𝑚𝑚𝑚𝑚 − 𝑛𝑛 21) 17 + 70 22) 3 18 2 2 𝑏𝑏 − 𝑏𝑏 𝑥𝑥 − 𝑥𝑥 − 23) 6 27 24) + 7 30 2 2 𝑎𝑎 − 𝑎𝑎 − 𝑝𝑝 𝑝𝑝 − 25) 15 + 50 26) + + 1 2 2 2 𝑚𝑚 − 𝑚𝑚𝑚𝑚 𝑛𝑛 𝑥𝑥 𝑥𝑥 27) + 10 + 16 28) 9 + 14 2 2 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑢𝑢 − 𝑢𝑢𝑢𝑢 𝑣𝑣 29) + 14 + 45 30) 5 45 + 40 2 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑛𝑛 − 𝑛𝑛 31) 5 + 20 25 32) 2 + 2 2 2 𝑣𝑣 𝑣𝑣 − 𝑚𝑚 − 𝑚𝑚𝑚𝑚 𝑛𝑛
252
Chapter 9 SECTION 9.3: FACTORING TRINOMIALS OF THE FORM ax2 + bx + c A. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c BY GROUPING When the leading coefficient a is not 1, it takes a few more steps to factor the trinomial. There are many ways to factor this type of trinomials. You are going to learn 2 methods in this section. The first one is factor by grouping and the second one is the “bottoms- up” method. First, let’s take a look at the grouping method.
Steps for factoring trinomials of the form ax2 + bx + c using the grouping method Step 1. Find two numbers, and , that + = and = Step 2. Rewrite the expression so that the middle term is split into two terms, p and q. 𝒑𝒑 𝒒𝒒 𝒑𝒑 𝒒𝒒 𝒃𝒃 𝒑𝒑 ⋅ 𝒒𝒒 𝒂𝒂 ⋅ 𝒄𝒄 Step 3. Factor by grouping. Step 4. ✓ Verify the factored form by finding the product.
MEDIA LESSON Factor trinomials when leading coefficient is not 1 - Grouping method (Duration 4:21)
View the video lesson, take notes and complete the problems below
a) 4 4 15 b) 20 + 19 + 3 ______2 ______2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 ______
YOU TRY
Factor using the grouping method and verify your answer by multiplying the two binomials. Show your work. a) 3 + 11 + 6 b) 8 2 15
2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −
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Chapter 9 B. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c BY THE “BOTTOMS UP” METHOD
Steps for factoring trinomials of the form ax2 + bx + c using the “bottoms-up” method Step 1. Multiply , then write a new trinomial in the form of x2 + bx + a∙c Step 2. Factor as you normally would with trinomials with the leading coefficient of 1. 𝒂𝒂 ⋅ 𝒄𝒄 Step 3. Divide the constants in each binomial factor by the original value of a. Step 4. Simplify the fractions formed. Step 5. If the simplified fractions does not have the denominator of 1, move the denominator to the coefficient of the variable. Step 6. ✓ Verify the factored form by finding the product
MEDIA LESSON Factor trinomials when the leading coefficient is NOT 1 - Bottoms up method (Duration 4:20)
View the video lesson, take notes and complete the problems below a) 4 4 15 b) 20 + 19 + 3 ______2 ______2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 ______
______
______
______
YOU TRY Factor the trinomials using the “bottoms up” method. Show your work. c) 3 + 11 + 6 d) 8 2 15 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −
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Chapter 9 C. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c BY THE TRIAL AND ERROR METHOD Factoring by trial-and-error is just a guess and check process when you try to add up different products to get the middle term bx. This sometimes works out faster than other methods above and sometimes not. If you want a step-by-step process that always works, this method may not be the best method for you.
MEDIA LESSON Factor trinomials when the leading coefficient is NOT 1 – Trial and error method (Duration 5:22)
View the video lesson, take notes and complete the problems below
a) 4 4 15 b) 20 + 19 + 3 2 2
(2 __𝑥𝑥____−____𝑥𝑥__−______)(2 ______) 𝑥𝑥 𝑥𝑥 (5 ______)(4 ______) ______𝑥𝑥 𝑥𝑥 ______𝑥𝑥 𝑥𝑥 ______(10 ______)(2 ______) (4 ______)(4 ______) ______𝑥𝑥 𝑥𝑥 ______𝑥𝑥 𝑥𝑥 ______(20 ______)( ______)
______𝑥𝑥 𝑥𝑥 ______(5 ______)(4 ______)
______𝑥𝑥 𝑥𝑥______(10 ______)(2 ______)
______𝑥𝑥 𝑥𝑥 (20 ______)( ______)
𝑥𝑥 𝑥𝑥 YOU TRY Factor the trinomials below by using the trial-error method:
a) 10 27 + 5 2 𝑥𝑥 − 𝑥𝑥
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Chapter 9 D. FACTORING TRINOMIALS OF THE FORM ax2 + bx + c WITH A GCF IN THE COEFFICIENTS As always, when factoring, we will first look for a GCF in the coefficients, factor the GCG, then factor the trinomial as usual.
MEDIA LESSON Factoring trinomials of the form ax2 + bx + c with GCF in the coeffients (Duration 1:45)
Stop at 7:00
View the video lesson, take notes and complete the problems below
Example: 18 15 12 2 𝑥𝑥 − 𝑥𝑥 −
YOU TRY a) 24 22 + 4 b) 18 33 + 30 2 3 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥
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Chapter 9 EXERCISE Factor completely by using grouping method. Show your work.
1) 7 48 + 36 2) 7 + 15 + 2 2 2 𝑥𝑥 − 𝑥𝑥 𝑏𝑏 𝑏𝑏 3) 5 13 28 4) 2 5 + 2 2 2 𝑎𝑎 − 𝑎𝑎 − 𝑥𝑥 − 𝑥𝑥 5) 2 3 6) 5 + 13 + 6 2 2 𝑏𝑏 − 𝑏𝑏 − 𝑘𝑘 𝑘𝑘 7) 3 17 + 20 8) 6 39 21 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 9) 6 29 + 20 10) 4 + 9 + 2 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 11) 4 + 13 + 3 12) 3 + 13 10 2 2 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑢𝑢 𝑢𝑢𝑢𝑢 − 𝑣𝑣
Factor completely by using the “bottoms up” method. Show your work.
13) 4 9 9 14) 4 + 13 + 3 2 2 𝑚𝑚 − 𝑚𝑚 − 𝑥𝑥 𝑥𝑥 15) 6 11 7 16) 4 + 3 2 2 𝑝𝑝 − 𝑝𝑝 − 𝑟𝑟 𝑟𝑟 − 17) 4 + 3 7 18) 3 4 4 2 2 𝑟𝑟 𝑟𝑟 − 𝑟𝑟 − 𝑟𝑟 − 19) 3 + 10 8 20) 2 5 3 2 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 21) 2 + 15 + 7 22) 7 11 + 4 2 2 𝑦𝑦 𝑦𝑦 𝑎𝑎 − 𝑎𝑎 24) 24 30 + 9 23) 4 + 16 + 16 2 2 𝑎𝑎 − 𝑎𝑎 𝑥𝑥 𝑥𝑥 25) 10 + 15 10 26) 2 + 12 + 18 3 2 3 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥 27) 5 + 15 + 10 28) 2 + 8 + 6 2 2 𝑡𝑡 𝑡𝑡 𝑥𝑥 𝑥𝑥 29) 7 2 5 30) 24 ^2 52 + 8 ^2 2 2 𝑥𝑥 − 𝑥𝑥𝑥𝑥 − 𝑦𝑦 𝑥𝑥 − 𝑥𝑥𝑥𝑥 𝑦𝑦 31) 4 + 13 + 3 32) 3 + 13 10 2 2 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑢𝑢 𝑢𝑢𝑢𝑢 − 𝑣𝑣
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Chapter 9 SECTION 9.4: SPECIAL PRODUCTS In the previous chapter, we recognized two special products: difference of two squares and perfect square trinomials. In this section, we discuss these special products to factor expressions.
A. DIFFERENCE OF TWO SQUARES When we see a binomial where both the 1st and 2nd terms are perfect square and one subtracts another, you have the difference of two squares. You can apply the following formula to factor quickly.
Difference of two squares = ( + )( ) 𝟐𝟐 𝟐𝟐 𝒂𝒂 − 𝒃𝒃 𝒂𝒂 𝒃𝒃 𝒂𝒂 − 𝒃𝒃 MEDIA LESSON Factor a Difference of Squares (Duration 4:19)
View the video lesson, take notes and complete the problems below
Example: Factoring binomials a) 36 b) 16 2 2 𝑥𝑥 − 𝑥𝑥 −
c) 100 9 d) 2 18 2 2 − 𝑥𝑥 𝑥𝑥 −
Warning: The sum of squares + does NOT factor. It is always prime. 𝟐𝟐 𝟐𝟐 𝒂𝒂 𝒃𝒃 MEDIA LESSON Factoring a difference of squares with two variables (Duration 1:48)
View the video lesson, take notes and complete the problems below
Factor: 49 2 2 𝑥𝑥 − 𝑦𝑦
YOU TRY
Factor completely: a) 9 b) 25 2 2 𝑥𝑥 − − 𝑚𝑚 c) 9 25 d) 5 45 2 2 2 2 𝑎𝑎 − 𝑏𝑏 𝑥𝑥 − 𝑦𝑦
258
Chapter 9 B. PERFECT SQUARE TRINOMIALS In this section, we discuss two special types of trinomials that are called the perfect square trinomials. In order to have a perfect square trinomial, we need to have the 1st and the 2nd terms squared and the middle term is twice the 1st and the 2nd terms. This pattern allows us to be more efficient when we factor trinomials. Perfect square trinomials + 2 + = ( + ) 2 2 + 2 = ( )2 𝑎𝑎2 𝑎𝑎𝑎𝑎 𝑏𝑏2 𝑎𝑎 𝑏𝑏 2 𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 MEDIA LESSON Factor perfect square trinomials (Duration 5:53)
View the video lesson, take notes and complete the problems below
Given a perfect square trinomial, factor the trinomial into the square of a binomial:
1. ______2. ______3. ______Example: 1. 36 + 60 + 25 2 𝑥𝑥 𝑥𝑥
2. 49 28 + 4 2 𝑥𝑥 − 𝑥𝑥
YOU TRY
Factor by using the perfect square formula: a) 6 + 9 b) 4 + 20 + 25 2 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦
259
Chapter 9 C. FACTORING SPECIAL PRODUCTS WITH A GCF IN THE COEFFICIENTS Sometimes, we have to factor a GCF out of a polynomial before we can apply the difference of two squares binomial or the perfect square trinomial formulas.
MEDIA LESSON Factor a difference of squares with a common factor (Duration 3:05)
View the video lesson, take notes and complete the problems below a) Example: 4 36 2 𝑥𝑥 −
YOU TRY
a) Factor completely: 72 8 b) Factor the GCF and apply the perfect square 2 formula: 3 18 + 27 𝑥𝑥 − 2 𝑥𝑥 − 𝑥𝑥
D. A SUM OR DIFFERENCE OF TWO CUBES Sum or difference of two cubes There are special formulas for a sum or difference of two cubes.
Difference of two cubes: = ( )( + + ) 3 3 2 2 Sum of two cubes: + 𝑎𝑎 −=𝑏𝑏 ( +𝑎𝑎 −)(𝑏𝑏 𝑎𝑎 𝑎𝑎+𝑎𝑎 )𝑏𝑏 3 3 2 2 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑏𝑏 We can also use the acronym SOAP for the formulas for factoring a sum or difference of two cubes.
Same: binomial has the same sign as the expression Opposite: middle term of the trinomial has the opposite sign than the expression Always Positive: last term of the trinomial is always positive
SOAP is an easier way of remembering the signs in the formula because the formulas for the sum and difference of two cubes are the same except for the signs. Let’s take a look:
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Chapter 9 MEDIA LESSON Factor a Sum or Difference of Cubes (Duration 4:58)
View the video lesson, take notes and complete the problems below
Special product – Cubes
Sum of cubes: ______
______
Difference of cubes: ______
______
Example: Factor. a) + 125 b) 8 27 3 3 3 𝑚𝑚 𝑎𝑎 − 𝑦𝑦
YOU TRY a) 27 b) 125 + 8 3 3 3 𝑚𝑚 − 𝑝𝑝 𝑟𝑟
Sometimes, you have to factor out the common factor in the coefficients before you can apply the Sum or Difference of Cubes formulas.
MEDIA LESSON Factor a Sum or Difference of Cubes when coefficients have common factor (Duration 3:34)
View the video lesson, take notes and complete the problems below
Factor completely: c) 4 32 d) 2 + 250 3 3 𝑥𝑥 − 𝑥𝑥
261
Chapter 9 MEDIA LESSON Factor a Sum or Difference of Cubes – Caution (Duration 5:22)
View the video lesson, take notes and complete the problems below
Example: Factor completely 64 + 216 . 6 3 64 216 𝑥𝑥 𝑦𝑦 /\ /\
YOU TRY a) 5 40 b) 128 + 54 3 4 2 5 𝑥𝑥 − 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑏𝑏
262
Chapter 9 EXERCISE Name the special product and factor completely. Show your work.
Name Name
1) ______2) 25 ______𝟐𝟐 2 𝒓𝒓 − 𝟏𝟏𝟏𝟏 𝑣𝑣 − 3) ______4) 9 4 ______𝟐𝟐 2 𝒑𝒑 − 𝟒𝟒 𝑘𝑘 − 5) ______6) 16 36 ______𝟐𝟐 2 𝟑𝟑𝒙𝒙 − 𝟐𝟐𝟐𝟐 𝑥𝑥 − 7) ______8) 2 + 1 ______𝟐𝟐 𝟐𝟐 2 𝟏𝟏𝟏𝟏𝒂𝒂 − 𝟐𝟐𝟐𝟐𝟐𝟐𝒃𝒃 𝑎𝑎 − 𝑎𝑎 9) + + ______10) 6 + 9 ______𝟐𝟐 2 𝒙𝒙 𝟔𝟔𝟔𝟔 𝟗𝟗 𝑥𝑥 − 𝑥𝑥 11) + + ______12) 4 + 4 ______𝟐𝟐 2 𝒌𝒌 𝟒𝟒𝟒𝟒 𝟒𝟒 𝑘𝑘 − 𝑘𝑘 13) + ______14) 8 24 + 18 ______𝟐𝟐 2 2 𝟐𝟐𝟐𝟐𝒑𝒑 − 𝟏𝟏𝟏𝟏𝟏𝟏 𝟏𝟏 𝑥𝑥 − 𝑥𝑥𝑥𝑥 𝑦𝑦 15) ______16) 64 ______𝟑𝟑 3 𝟖𝟖 − 𝒎𝒎 𝑥𝑥 − 17) ______18) 125 64 ______𝟑𝟑 3 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝒖𝒖 𝑎𝑎 − 19) + ______20) + 64 ______𝟑𝟑 𝟑𝟑 3 𝒙𝒙 𝟐𝟐𝟐𝟐𝒚𝒚 𝑥𝑥 21) ______22) 1 ______𝟑𝟑 𝟑𝟑 2 𝟑𝟑𝟑𝟑𝒎𝒎 − 𝟒𝟒𝟒𝟒 𝑥𝑥 − 23) + ______24) 9 1 ______𝟐𝟐 2 𝟒𝟒𝒂𝒂 𝟐𝟐− 𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐 𝑎𝑎 − 25) 𝟐𝟐𝟐𝟐𝒃𝒃 ______26) 125 + 27 ______𝟐𝟐 3 3 𝒙𝒙 − 𝟗𝟗 𝑥𝑥 𝑦𝑦 27) ______28) 25 + 30 + 9 ______𝟐𝟐 2 2 𝟒𝟒𝒗𝒗 − 𝟏𝟏 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏 29) ______30) + 8 + 16 ______𝟐𝟐 2 2 𝟓𝟓𝒏𝒏 − 𝟐𝟐𝟐𝟐 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 31) ______32) 20 + 20 + 5 ______𝟐𝟐 𝟐𝟐 2 2 𝟒𝟒𝒎𝒎 − 𝟔𝟔𝟔𝟔𝒏𝒏 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 33) + ______34) + 8 ______𝟐𝟐 3 𝒏𝒏 − 𝟖𝟖𝟖𝟖 𝟏𝟏𝟏𝟏 𝑥𝑥 35) + + ______36) 64 27 ______𝟐𝟐 3 𝒙𝒙 𝟐𝟐𝟐𝟐 𝟏𝟏 𝑥𝑥 − 37) + ______38) 18 24 + ______𝟑𝟑 8 2 𝟏𝟏𝟏𝟏𝟏𝟏𝒙𝒙 𝟐𝟐𝟐𝟐𝟐𝟐 𝑚𝑚2 − 𝑚𝑚𝑚𝑚 39) ______40) 𝑛𝑛 16 ______𝟒𝟒 𝟒𝟒 4 𝒙𝒙 − 𝒚𝒚 𝑧𝑧 −
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Chapter 9
SECTION 9.5: FACTORING, A GENERAL STRATEGY A General Strategy To Factoring Step 1. Factor out the greatest common factor, if possible. Step 2. Determine the number of terms in the polynomial. Step 3. a) Two Terms • Difference of two squares: = ( + )( ) • Difference of two cubes: 2 2= ( )( + + ) 𝑎𝑎 − 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 • Sum of two cubes: + 3 = (3 + )( 2 + ) 2 𝑎𝑎 − 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏 b) Three Terms 3 3 2 2 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑏𝑏 • Perfect square trinomial: + 2 + = ( + ) or 2 + = ( ) • Old fashion way: 2 2 2 2 2 2 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 . + + = ( + )( + ): using the ac method . 2 + + : Factor by grouping or by the “bottoms up” method. c) Four Terms 𝑥𝑥 2 𝑏𝑏𝑏𝑏 𝑐𝑐 𝑥𝑥 𝑝𝑝 𝑥𝑥 𝑞𝑞 • Factor by grouping,𝑎𝑎𝑎𝑎 rearranging𝑏𝑏𝑏𝑏 𝑐𝑐 terms, if needed.
Step 4. Check your work by multiplying out the product of factors.
MEDIA LESSON General factoring strategy (Duration 5:00)
View the video lesson, take notes and complete the problems below
• Always do ______first! • 2 terms: 3 terms: 4 terms:
Example: a) 25 16 b) 20 c) + 2 + 5 + 10 ______2 ______2 ______𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑥𝑥
264
Chapter 9 EXERCISE Apply the general factoring strategy to factor the following polynomials completely. Show your work.
1) 24 18 + 60 45 2) 2 128 3 3 𝑎𝑎𝑎𝑎 − 𝑎𝑎ℎ 𝑦𝑦𝑦𝑦 − 𝑦𝑦ℎ 𝑥𝑥 − 𝑦𝑦
3) 54 16 4) 4 + 3 3 2 2 𝑢𝑢 − 𝑥𝑥 − 𝑥𝑥𝑥𝑥 𝑦𝑦
5) 4 6) 128 + 54 2 2 3 𝑚𝑚 − 𝑛𝑛 𝑥𝑥
7) + 7 + 10 8) 5 + 2 3 2 2 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑥𝑥 𝑥𝑥
9) 12 + 3 4 10) 27 48 2 2 𝑚𝑚𝑚𝑚 − 𝑥𝑥 𝑚𝑚 − 𝑥𝑥𝑥𝑥 𝑚𝑚 − 𝑛𝑛
11) 16 + 48 + 36 12) 2 + 5 + 3 2 2 3 2 2 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑥𝑥
13) 5 22 15 14) 27 2 3 3 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦
15) 3 6 24 16) 3 + 15 + + 5 3 2 2 2 2 2 2 𝑚𝑚 − 𝑚𝑚 𝑛𝑛 − 𝑛𝑛 𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑑𝑑 𝑥𝑥 𝑐𝑐 𝑥𝑥 𝑑𝑑
17) 16 9 18) 32 18 2 2 2 2 𝑎𝑎 − 𝑏𝑏 𝑥𝑥 − 𝑦𝑦
19) 20) 9 3 2 3 2 𝑣𝑣 − 𝑣𝑣 𝑛𝑛 − 𝑛𝑛
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Chapter 9 SECTION 9.6: SOLVE BY FACTORING When solving linear equations, such as 2x − 5 = 21, we can solve by isolating the variable on one side and a number on the other side. However, in this chapter, we have an x2 term, so if it looks different, then it is different. Hence, we need a new method for solving trinomial equations. One method is using the zero product rule. There are other methods for solving trinomial equations, but that is for a future chapter.
Definition A polynomial equation is any equation that contains a polynomial expression. A trinomial equation is written in the form + + =
𝟐𝟐 where a, b, c are coefficients, and a ≠𝒂𝒂 0.𝒂𝒂 If the𝒃𝒃 𝒃𝒃trinomial𝒄𝒄 𝟎𝟎equations have the highest power is 2, they are also called as quadratic equations.
A. ZERO PRODUCT RULE
Zero product rule If ; are non-zero factors, then = implies = or = or both = = .
𝒂𝒂 𝒃𝒃 𝒂𝒂 ∙ 𝒃𝒃 𝟎𝟎 𝒂𝒂 𝟎𝟎 𝒃𝒃 𝟎𝟎 𝒂𝒂 𝒃𝒃 𝟎𝟎 MEDIA LESSON Solve equations by using the Zero product rule (Duration 4:04) )
View the video lesson, take notes and complete the problems below
Zero product rule: ______To solve we set each ______equal to zero
Example: a) (5 1)(2 + 5) = 0 b) 2 ( 6)(2 + 3) = 0
𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
YOU TRY
Solve for x:
a) ( + 7) = 0 b) (2x 3)(5x + 1) = 0
𝑥𝑥 𝑥𝑥 −
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Chapter 9 B. SOLVE EQUATIONS BY FACTORING
Steps for solving trinomial equations
Step 1. Write the given equation in the form + + = 0. Step 2. Factor the left side of the equation into a2 product of factors. 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐 Step 3. Use the zero product rule to set each factor equal to zero and then solve for the unknown. Step 4. Verify the solution(s).
MEDIA LESSON Solve quadratic equations by factoring the GCF (Duration 2:38)
View the video lesson, take notes and complete the problems below
Example: Solve a) + 4 = 0 b) 14 35 = 0 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
YOU TRY Solve the equations by factoring:
a) 9 = 0 b) 7 28 = 0 2 2 𝑛𝑛 − 𝑛𝑛 𝑛𝑛 − 𝑛𝑛
MEDIA LESSON Factor and solve quadratic equations when a = 1 (Duration 5:24 )
View the video lesson, take notes and complete the problems below
Example: Solve a) + 6 + 8 = 0 b) + 23 50 = 0 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 −
c) 8 + 15 = 0 d) 5 24 = 0 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −
267
Chapter 9 YOU TRY Solve the equations by factoring a) + 5 + 6 = 0 b) 9 + 14 = 0 2 2 𝑐𝑐 𝑐𝑐 𝑦𝑦 − 𝑦𝑦
MEDIA LESSON Factor and solve quadratic equation with a negative leading coefficient (Duration 5:10)
View the video lesson, take notes and complete the problems below
Example: Solve by factoring. a) + 7 + 18 = 0 b) 12 36 = 0 2 2 −𝑥𝑥 𝑥𝑥 −𝑥𝑥 − 𝑥𝑥 −
YOU TRY
Solve by factoring: a) + + 6 = 0 b) 3 + 18 = 0 2 2 −𝑥𝑥 𝑥𝑥 −𝑦𝑦 − 𝑦𝑦
268
Chapter 9 MEDIA LESSON Factor & solve quadratic equations with common factor in the coefficients (Duration 4:27)
View the video lesson, take notes and complete the problems below
Example: Solve by factoring. a) 3 + 15 + 18 = 0 b) 8 72 + 162 = 0 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
YOU TRY
Solve by factoring: c) 3 24 + 45 = 0 d) 4 + 52 + 168 = 0 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥
MEDIA LESSON Factor and solve a quadratic equation when the leading coefficient is NOT 1 (Duration 6:17)
View the video lesson, take notes and complete the problems below
Example: Solve by factoring. a) 4 + 25 21 = 0 b) 3 23 + 30 = 0 2 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥
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Chapter 9 YOU TRY Solve by factoring a) 3 + 11 + 6 = 0 b) 8 2 15 = 0 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −
MEDIA LESSON Factor and solve a quadratic equation using the difference of 2 squares (Duration 3:58)
View the video lesson, take notes and complete the problems below
Example: Solve by factoring a) 49 = 0 b) 4 = 81 2 2 𝑥𝑥 − 𝑥𝑥
MEDIA LESSON Factor and solve a quadratic equation using the difference of 2 squares with GCF (Duration 5:03)
View the video lesson, take notes and complete the problems below
Example: Solve by factoring c) 48 75 = 0 d) 2 = 32 2 2 𝑥𝑥 − 𝑥𝑥
270
Chapter 9 YOU TRY Factor binomials and solve the equation a) 9 = 0 b) 8 = 50 2 2 𝑥𝑥 − 𝑥𝑥
MEDIA LESSON Factor and solve a quadratic equation using the perfect square trinomials (Duration 4:52 )
View the video lesson, take notes and complete the problems below
Example: Solve by factoring. a) + 4 + 4 = 0 b) 10 + 25 = 0 2 2 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
c) 4 12 + 9 = 0 2 𝑥𝑥 − 𝑥𝑥
YOU TRY
Solve by factoring: a) 2 + 1 = 0 b) 9 + 6 + 1 = 0 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥
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Chapter 9 C. SIMPLIFY THE EQUATION Sometimes the equation isn’t so straightforward. We may have to do some preliminary work so that the equation takes the form of a trinomial equation and then we can use the zero product rule.
MEDIA LESSON Solve by factoring – Simplify first (Duration 4:57)
View the video lesson, take notes and complete the problems below
Example:
a) 2 ( + 4) = 3 3 b) (2 3)(3 + 1) = 8 1
𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 −
YOU TRY Simplify the following equations and solve by factoring.
a) ( 7)( + 3) = 9 3 + 4 5 = 7 + 4 14 2 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 −
272
Chapter 9 EXERCISE Solve each equation by factoring. Show your work.
1) ( 7)( + 2) = 0 2) ( 1)( + 4) = 0
𝑘𝑘 − 𝑘𝑘 𝑥𝑥 − 𝑥𝑥 3) 6 150 = 0 4) 2 + 10 28 = 0 2 2 𝑥𝑥 − 𝑛𝑛 𝑛𝑛 − 5) 7 + 26 + 15 = 0 6) 5 9 2 = 0 2 2 𝑥𝑥 𝑥𝑥 𝑛𝑛 − 𝑛𝑛 − 7) 4 8 = 8 8) 5 1 = 5 2 2 𝑥𝑥 − 𝑥𝑥 − − 𝑥𝑥 − 𝑥𝑥 − − 9) 49 + 371 163 = 5 10) 7 + 17 20 = 8 2 2 𝑝𝑝 𝑝𝑝 − 𝑥𝑥 𝑥𝑥 − − 11) 7 + 84 = 49 12) 6 = 16 2 2 𝑟𝑟 − 𝑟𝑟 𝑥𝑥 − 𝑥𝑥 13) 3 + 7 = 40 14) 35 + 120 = 45 2 2 𝑣𝑣 𝑣𝑣 𝑥𝑥 𝑥𝑥 − 15) 4 + 18 23 = 6 7 16) 9 46 + 7 = 7 + 8 + 3 2 2 2 𝑘𝑘 𝑘𝑘 − 𝑘𝑘 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 17) 2 + 19 + 40 = 2 18) 40 + 183 168 = + 5 2 2 2 𝑚𝑚 𝑚𝑚 − 𝑚𝑚 𝑝𝑝 𝑝𝑝 − 𝑝𝑝 𝑝𝑝 19) ( + 4)( 3) = 0 20) (2 + 5)( 7) = 0
𝑎𝑎 𝑎𝑎 − 𝑥𝑥 𝑥𝑥 − 21) + 4 32 = 0 22) 30 = 0 2 2 𝑝𝑝 𝑝𝑝 − 𝑚𝑚 − 𝑚𝑚 − 23) 40 285 280 = 0 24) 2 3 2 = 0 2 2 𝑟𝑟 − 𝑟𝑟 − 𝑏𝑏 − 𝑏𝑏 − 25) 8 3 = 3 26) 6 + 6 = 2 2 2 𝑣𝑣 − 𝑣𝑣 − − 𝑎𝑎 − 𝑎𝑎 − 27) 7 + 57 + 13 = 5 28) 7 28 = 0 2 2 𝑘𝑘 𝑘𝑘 𝑛𝑛 − 𝑛𝑛 29) 6 = 5 + 7 30) 9 + 39 = 36 2 2 𝑏𝑏 𝑏𝑏 𝑛𝑛 𝑛𝑛 − 31) + 7 9 = 3 + 6 32) + 10 + 30 = 6 2 2 𝑎𝑎 𝑎𝑎 − − 𝑎𝑎 𝑥𝑥 𝑥𝑥 33) 5 + 41 + 40 = 2 34) 24 + 11 80 = 3 2 2 𝑛𝑛 𝑛𝑛 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥
273
Chapter 9 CHAPTER REVIEW KEY TERMS AND CONCEPTS Look for the following terms and concepts as you work through the workbook. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the text or in the media lesson.
Factored form
Greatest common factor (GCF)
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