Algebra 1 Monomials/Polynomials/Factoring Packet
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Algebra 1 Monomials/Polynomials/Factoring packet
Multiplying and Dividing Monomials
Multiplying:
32 = 3 3 = 9 4 4 = 42 = 1643 = (4) (4) (4) = 64(5)(5)(5)(5)(5)(5) = 56=15,625
The same goes for variables:
x x = x2
x2 x3 = (x)(x) (x)(x)(x) = x5
(The only difference is you can’t simplify x2 like you did 32 = 9. You must leave it as x2.)
When multiplying monomials you must deal with the coefficients.
Coefficients:Multiply the coefficients.
Variables:When multiplying the variables of monomials you keep the base and add the exponents. (Remember if there is no exponent written, the exponent is 1.)
Look at the previous example:x1 x1 = x(1+1) = x2
Simplify: (3xy5)(4x2y3)
(3xy5)(4x2y3) = (3)(4)(x)( x2)(y5)(y3) = 12 [x(1+2) ][y(5+3)] = 12x3y8
Dividing:
64/62 = (6)(6)(6)(6) cancel (6)(6) (6)(6) = (6)(6) = 62= 36
(6)(6) (6)(6)
x3/x = (x)(x)(x) cancel (x)(x) (x) = (x)(x) = x2
(x) (x)
Just like multiplying, when dividing monomials you must deal with the coefficients.
Coefficients : Divide the coefficients.
Variables: When dividing the variables of monomials you keep the base and subtract the exponents.
Look at the previous example: x3/x = x3-1 = x2
Simplify: (12xy5)/(4xy3) =
12/4 = 3 x1-1 = x0 y5-3 = y2
What is x0 equal to? : Any number or variable with an exponent of 0 = ?
Final answer =
Do all examples in NB.
Show all steps!
1) Multiply:
a) (5x3y2z11)(12xy7z-4)b) (9x5y2z4)3c) (4x3y7z6)4(3x8y -5z -12)2
2) Multiply:
a) (6x3y2z-12)(11x5y-3 z7)b) (8x5y-2z4)4c) (3x6y5z8)3(5x-9y 5z -15)2
3) Divide:
a) 27x3y2z5 .b) (4x4y5z)3c) (2x5yz6)5
9x3y5z4 16x4y13z4 (4x11y5z14)2
4) Divide:
a) 45x3y9z5 .b) 24x8y12z9 c) 32x5y12z28
18x6y5z 72x10y12z8 8x7y-12z14
5) (3x5y8z5)5 6) (6x5y4z6)3
(9x14y20z12)2 (12x7y8z-9)2
7) 5a(8a2 – 6a + 3) – 3a(11a2 – 10a – 5)8) 8b(7b2 – 4b + 2) – 5(6b2 + 3b – 1)
9)7x(4x2 - 11x + 3) - 4x(8x2 -18x + 5)10) 5x(7x2 - 6x + 4) - 3x(10x2 -7x- 1)
11) 6y2(5y3 – 4y2 + 8y – 7) – 8y(3y3 + 6y2 – 5y – 9)
When MULTIPLYING monomials you the coefficients and the exponents.
When DIVIDING monomials you the coefficients and
the exponents.
1) (3x9y)(6x11y4)2) 36x9y6z5 _3) (7x2yz3)3
12x-9y6z4
4) 45x4y3z7 _5) (4x5yz3)3 6) (5x2y2z-4)(2x-5y3z)3
18x6y-3z5 (2x3y6z-2)5
7) (6x7y4z3)2(2x-5y3z)38) (9x2y5z-11)2 _9) (6x2y5z3)2 _
(3x-2y2z4)5 (2x-3y2z2)5
10) 4x(9x2 - 15x - 12) - 12x(3x2 + 5x- 4)11) 3y2(5y3 – 4y2 + 8y – 7) – 7y(3y3 + 6y2 – 5y – 9)
Multiplying binomials:
We have a special way of remembering how to multiply binomials called FOIL:
F:firstx x = x2(x + 7)(x + 5)
O:outerx 5 = 5x
I:inner7 x = 7xx2 + 5x +7x + 35 (then simplify)
L:last7 5 = 35x2 + 12x + 35
1) (x - 5)(x + 4)2) (x - 6)(x - 3) 3) (x + 4)(x + 7) 4) (x + 3)(x - 7)
5) (3x - 5)(2x + 8)6) (11x - 7)(5x + 3)7) (4x - 9)(9x + 4) 8)(x - 2)(x + 2)
9) (x - 2)(x - 2)10)(x - 2)2 11) (5x - 4) 2 12) (3x + 2)2
Factoring using GCF:
Take the greatest common factor (GCF) for the numerical coefficient. When choosing the GCF for the variables, if all the terms have a common variable, take the one with the lowest exponent.
ie) 9x4 + 3x3 + 12x2
GCF: coefficients:3
Variable (x) :x2
GCF: 3x2
What’s left? Division of monomials:
9x4/3x23x3 /3x212x2/3x2
3x2 x 4
Factored Completely:3x2 (3x2 + x+ 4)
Factor each problem using the GCF and check by distributing:
1) 14x9 - 7x7 + 21x52) 26x4y - 39x3y2 + 52x2y3 - 13xy4
3) 32x6 - 12x5 - 16x44) 16x5y2 - 8x4y3 + 24x2y4 - 32xy5
5) 24b11 + 4b10 -6b9 + 2b86) 96a5b + 48a3b3 - 144ab5
7) 11x3y3 + 121x2y2 - 88xy8) 75x5 + 15x4 -25x3
9) 132a5b4c3 - 48a4b4c4 + 72a3b4c510) 16x 5+ 12xy - 9y5
HOW TO FACTOR TRINOMIALS
Remember your hints:
A. When the last sign is additionB. When the last sign is subtraction
x2 - 5x + 61)Both signs the samex2 + 5x – 361) signs are different
2) Both minus (1st sign)
(x - )(x - )(x - )(x + ) 2) Factors of 36 w/ a
3) Factors of 6 w/ a sum differenceof 5 (9
of 5. (3 and 2) and 4)
3) Bigger # goes1st sign,+
(x - 3)(x - 2)(x - 4)(x + 9)
FOIL Check!!!!!
Factor each trinomial into two binomials check by using FOIL:
1) x2 + 7x + 62) t2 – 8t + 123) g2 – 10g + 16
4) r2 + 4r - 215) d2 – 8d - 336) b2 + 5b - 6
7) m2 + 16m + 648) z2 + 11z - 269) f2 – 12f + 27
10) x2 - 17x + 7211) y2 + 6y - 7212) c2 + 5c - 66
13) z2 – 17z + 5214) q2 – 22q + 12115) w2 + 8w + 16
16) u2 + 6u - 717) j2 – 11j - 4218) n2 + 24n + 144
19) t2 + 2t -3520) d2 – 5d - 6621) r2 – 14r + 48
22) p2 + p - 4223) s2 + s - 5624) b2 – 14b + 45
25) f2 + 15f + 3626) n2 + 7n - 1827) z2 + 10z - 24
28) h2 + 13h + 2429) w2 + 29w + 2830) v2 – 3v – 18
31) y2 - 932) g2 – 3633) t2 – 121
34) 9k2 – 2535) 144m2 – 4936) 64e2 – 81
37) a2 + 10038) w2 – 4439) d2 – d– 9
Factor using GCF and then factor the trinomial (then check):
40) 4b2 + 20b + 2441) 10t2 – 80t + 15042) 9r2 + 90r - 99
43) 3g3 + 27g2 + 60g44) 12x6 + 72x5 + 60x445) 8c9 + 40c8 - 192c7
46) 12d2 – 1247) 25r2 – 10048) 5z5 – 320z3
Case II Factoring
Factoring a trinomial with a coefficient for x2 other than 1
Factor:6x2 + 5x – 4
1)Look for a GCF:
a.There is no GCF for this trinomial
b.The only way this method works is if you take out the GCF (if there is one.)
2)Take the coefficient for x2 (6) and multiply it with the last term (4):
6x2 + 5x – 46 * 4 = 24
x2 + 5x – 24
3)Factor the new trinomial:
x2 + 5x – 24
(x + 8)(x – 3)
4)Take the coefficient that you multiplied in the beginning (6) and put it back in the parenthesis (only with the x):
(x + 8)(x – 3)
(6x + 8)(6x – 3)
5)Find the GCF on each factor (on each set of parenthesis):
(6x + 8) 2(3x+ 4)
(6x – 3) 3(2x – 1)
6)Keep the factors left in the parenthesis:
(3x + 4)(2x – 1)
7)FOIL CHECKExtra Problems: (Remember... GCF 1st)
1) 7x2 + 19x – 6
(3x + 4)(2x – 1)2) 36x2 - 21x + 3
3) 12x2 - 16x + 5
6x2 –3x + 8x – 44) 20x2 +42x – 20
5) 9x2 - 3x – 42
6x2 + 5x – 46) 16x2 - 10x + 1
7) 24x2 + x – 3
8) 9x2 + 35x – 4
9) 16x2 + 8x + 1
10) 48x2 + 16x – 20
Factor each trinomial and FOIL Check:
1) x2 – 6x – 722) x2 + 14x + 133) x2 – 19x + 88
4) x2 + 2x – 635) x2 – 1966) x2 – 1
7) x2 + 20x + 648) x2 + 11x - 129) x2 - 12x + 35
10) x2 - 17x + 7011) x2 + 14x - 7212) x2 + 5x – 36
13) x2 - 20x + 9614) x2 - 24x + 14415) x2 + 10x + 25
Factor using the GCF:
16) 24x10 - 144x9 + 48x817) 64x5y3 – 40x4y4 + 32x3y4 – 8x2y3
Factor using the GCF and then factor the quadratic:
18) x4 – 15x3 + 56x219) 4x2 + 24x – 24020) 5x3 – 5x2 –360x
21) 12x2 – 24322) 16x2 – 1623) 8x17 – 512x15
Mixed Problems:
24) 49x2 – 2525) 4x2 – 12126) x4 – 36
27) x16 – 6428) x100 – 16929) 48x8 – 12
30) 25x2 – 10031) 36x4 – 932) 100x2 – 225
33) x2 + 6434) x2 – 4835) x2 – 2x + 24
36) x2 + 11x – 3037) 5x2 + 2038) 7x2 – 7x - 84
1-Step Factoring:Factor each quadratic. If the quadratic is unable to be factored, your answer should be PRIME.
Examples:
(last sign +)(last sign - )(D.O.T.S)
x2 – 10x + 24x2 + x – 12x2 – 49
Same sign, both -Different SignsDiff of Two Sq.
Factors of 24, sum of 10Factors of 12, diff. of 1
(x – 6)(x – 4)(x + 4)(x – 3)(x + 7)(x – 7)
1) x2 + 5x + 42) a2 – 12a + 353) f2 – 3f – 18
4) g2 + 5g – 505) t2 – 2t + 486) x2 – 100
7) s2 – 9s + 208) j2 + 7j + 129) k2+ 2k – 24
10) x2 – 6x – 711) n2 -2512) c2 – 13c – 40
13) g2 – 5g – 8414) z2 + 17z + 7215) q2– 3q + 18
16) p2– 8117) w2 – w – 13218) x2 + 13x – 48
19) z2 + 9z – 3620) h2 + 12h + 3621) r2 + 5r + 36
22) b2 – 5b – 3623) x2 – 3624) m2 – 20m + 36
25) y2 – 4y – 6026) v2 + 16v – 6027) r2 + 7r – 60
28) x2 + 61x + 6029) g2 – 23g + 6030) b2 – 121
31) a2 + 4a – 9632) y2 – y – 11033) x2 + x + 90
34) t2 + 21t + 10835) w2 – 6436) x2 – 14x + 49
2-Step Factoring: Factor using the GCF and then try to factor what’s left.
Example:6x2 – 18x + 12
6(x2 – 3x + 2)
6(x – 2)(x – 1)
37) 5x2 + 10x - 12038) 3w2 -33w +9039) 8t2 – 32t - 256
40) 6d2 + 60d + 15041) 9x2 - 3642) 10z2 + 50z - 240
43) 7f2 + 84f + 25244) 2x2 – 2x - 18045) 4s2 - 144
46) 5g2 - 24547) 9k2 – 99k + 25248) 25k2 – 225
Case II: Factor using your steps for Case II factoring. Remember GCF is always the 1st step of any type of factoring!!!
Example: 6x2 – 5x – 4 (mult. 1st by last)
x2 – 5x – 24 (factor)
(x – 8)(x + 3)(put the 1st #, 6, back in)
(6x – 8)(6x + 3) (reduce: take 2 out of the 1st factor
and 3 out of the 2nd)
(3x – 4)(2x + 1)
49) 2x2 – 7x - 3050) 12s2 + 19s + 451) 18c2 + 9c - 2
52) 18y2 + 19y + 553) 15f2 – 14f + 354) 15k2 + 7k - 8
55) 12s2 – 22s - 2056) 24d2 – 6d - 3057) 21w2 + 93w + 36
58) 40x2 + 205x + 2559) 100z2 + 10z - 2060) 24r2 – 90r + 21
Factoring with 2 Variables
You will follow all your same factoring rules.
1) Look for a GCF.
2) Choose whether it is Case I, DOTS, or CASE II and proceed.
Remember: All the rules of factoring quadratics come from what will result when you FOIL check the two binomials:
Example:x2 – xy – 6y2
Usually, we start off with (x + )(x - ). We do this because we know when we FOIL, with the F (first), we need x ∙ x to get us x2. Then we follow the rest of the rules.
This one is just a little different. Now we’ll start with (x + __ y)(x - __ y), because not only do we need the x ∙ x to get x2, we’ll need a y ∙ y to get us the y2 we need at the end of the trinomial when we do the L (last).
Once you have that you can proceed. We will need factors of 6 with a difference of 1. The factors of 3 and 2 satisfy that and we will put the 3 with the minus sign because that is where the 1st sign tells us to put the bigger #. (x + 2y)(x - 3y)
Examples:
1) x2 – 11xy + 28y22) c2 + 3cd – 54d23) s2 – 25t2
4) 3u2 + 36uv+ 105v25) 5a2 – 25ab – 180b26) 49f 2 – 25g2
7) 7r2 – 175s28) 36x2 – 9y29) 225w2 – 100x2