Download Factoring Polynomials

7.1 FACTORING POLYNOMIALS
The greatest common factor (GCF) of an expression consists of the largest monomial that
divides (is common to) all terms and any variables that appear in every term with an
exponent equal to the smallest exponent of that variable in the expression.
Factor out common factors: For numbers, look for a positive or negative number that divides evenly into
all terms; for variables, if the same variable appears in all terms, factor out that variable using the smallest exponent.
1. 4a2b2 + 12ab3 + 4ab2
2. -13y8 + 26y4 – 39y2
3. 5(x + y) + a(x + y)
4. 2x2(x + 2) – 5x (x + 2) + 9(x + 2)
Factor by grouping:
If polynomial has four or more terms, separate it into two or more groups, factor any
common factor out of each group and look for a common binomial factor. Remember that 1 and -1 are factors of anything.
5. 4x3 + 3x2y + 4xy2 + 3y3
6. 18r2 + 12ry – 3ry – 2y2
To factor a trinomial, x2 + bx + c, find factors of c whose sum = b. Check first for
common factors; then factor by trial & error. When constant term c is positive, signs in
the binomials will be the same and will match the middle term. When constant term c is
negative, signs in the binomials will be different.
1. a2 + 9a + 20
2. b2 - 8b + 15
Look for factors of –42
whose sum is -1.
Look for factors of 20
whose sum is 9.
4. n2 – 2n – 35
3. q2 – q – 42
5. 3y3z + 9y2z2 – 162yz3
6. 2m3n – 20m2n2 + 24mn3
To factor a trinomial, ax2 + bx + c, find factors of a and c where the sum of the inner
and outer products = b; e.g., (a1 + c1)(a2 + c2) where a1c2+a2c1 = b. Check to see that you
have factored completely.
7. 4x2 + 11x + 6
10. 3a2 + 6a + 3
8. 30a2 – 38a + 12
11. x3 – 5x2 – 6x
9. 7u2 + 11u - 6
12. 3x2(x + 5) – 19x(x + 5) – 14(x + 5)
Special Factoring
Differences of two squares: a2 – b2 = (a + b)(a – b)
1. 4x2 – 9
2. 9r2 – 1
3. 16y2 + 25
4. 100b2 – 4/49
5. 36m2 – 16/25
6. 32a3 – 8ab2
7. 36x5 – 16xy2
8. 16k4 – 1
9. y4 - 16
Perfect Square Trinomials: a2 + 2ab + b2 = (a + b)2
and a2 – 2ab + b2 = (a – b)2
10. 9x2 – 42xy + 49y2
11. 16x2 + 40x + 25
12. 3y2 – 48y + 192
13. 2x2 + 24x + 72
14. -18x2 – 48xy – 32y2
15. -50h2 + 40hy – 8y2
Difference of two cubes: a3 – b3 = (a – b)(a2 + ab + b2)
Sum of two cubes: a3 + b3 = (a + b)(a2 – ab + b2)
16. m3 + 8
17. a3 - 1
18. 64x3 – 125y3
20. 3x4y – 15x2y – 108y
19. 27a3 + 343b3