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Polynomial factoring guide.
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1
FACTORING POLYNOMIALS
Factoring refers to the process of expressing an algebraic expression as a product of prime factors. A factor
is considered prime if it could no longer be expressed further as a product of simpler algebraic expressions, i.e.
its only factors are 1 and itself. An algebraic expression that is expressed as a product of irreducible factors is
factored completely.
Types of Factoring
1. Common Monomial Factor
ax + ay = a (x + y)
2. Difference of Two Squares
x2 − y2 = (x + y) (x− y)
3. Perfect Square Trinomial
x2 + 2xy + y2 = (x + y)2
x2 − 2xy + y2 = (x− y)2
4. Sum and Difference of Two Cubes
x3 + y3 = (x + y) (x2 − xy + y2)
x3 − y3 = (x− y) (x2 + xy + y2)
5. Quadratic Trinomials
x2 + (a + b) x + ab = (x + a) (x + b)
acx2 + (ad + bc) x + bd = (ax + b) (cx + d)
6. Completing the Square
This technique is applicable to polynomials that may be converted to a perfect square trinomial upon addition
and then subtraction of a perfect square term.
7. Factoring by Grouping
This technique is applicable to polynomials that are longer or more complicated. The key lies in grouping
the terms in such a way that the groups have common factor. This may entail several trials before the desired
grouping is arrived at.
EXERCISES
1. −12xy3z2 − 28y3z − 20x2y2z2
2. 4ab3 − 16a3b
3. 4 (x− 1)2 − 9y2
4. (2x− y)3 − 8
5. (x + 3)3 + (y − 1)3
6. 16a2b4 + 40ab2c + 25c2
7. (2s− 3t)2 − 8 (2s− 3t) + 16
8. 5x3 − 10x2y − 75xy2
9. x (x + 1) (4x− 5)− 6 (x + 1)
10. a2 − ab + a− b
11. x2 + xy − 2y2 + 2x− 2y
12. 4x2 − y2 + 2yz − z2
13. 9x2 − 12xy + 4y2 − 25z2 + 10zw − w2
14. x4 + 64
15. x4 − 11x2 + 1
16. 16x4 − 24x2y2 + 25y2
mong!