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 ................................................................................................................................................. 274 241 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, but backwards − . We will start− with 8 12 + 32 and obtain its factored form. 2 3 − 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 88 24 /\ /\ 88 = _________________________________ 18 24 = _________________________________ 13 GCF = _________________________________ YOU TRY Find the GCF a) 24 and 56 b) 12 , 6 and 21 3 15 5 20 7 242 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 /\ /\ 108 = __________________________________ 5 3 96 __________________________________ 7 2 GCF=__________________________________ YOU TRY Find the GCF: a) 15 and 45 b) 24 , 18 , and 12 3 6 4 2 4 2 2 4 3 5 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 − − YOU TRY Find the GCF: a) 4 20 + 10 b) 6 15 + 9 2 4 3 2 − − 243 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 /\ /\ /\ GCF = ____________ _____________________________________________________________________________________ b) 60 15 + 45 604 3 3 4 2 5 15 48 /\ − /\ /\ GCF = ______________ _____________________________________________________________________________________ _____________________________________________________________________________________ YOU TRY Factor: a) 25 15 + 20 b) 12 20 16 4 6 3 3 2 4 3 5 4 − − − 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 − − YOU TRY Factor using the division method. a) 21 + 14 + 7 b) 4 20 + 16 3 2 2 − 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 − − − 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) − − − YOU TRY Factor: a) 3 (2 + 5 ) 4 (2 + 5 ) b) (9 2)3 (9 2)5 2 2 − − − − 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