Algebra (COMPLETE FACTORIZATION)

Factor a^3 - 3 + 3a^2 - a.

(a - 1)(a + 1)(a + 3)

Factor completely. m^4 - n^4

(m^2 + n^2)(m + n)(m - n)

Factor completely. n^4 - 1

(n^2 + 1)(n - 1)(n + 1)

Factor completely. x^8 - 1/81

(x^4 + 1/9)(x^2 + 1/3)(x^2 - 1/3)

Factor z^3 + 2z^2 - 9z - 18

(z - 3)(z + 3)(z + 2)

Which of the following is the complete factorization of 10x - 3 - 3x^2

-(3x - 1)(x - 3)

Factor -4 + 8a - 4a^2

-4(a - 1)^2

Given: -7y^2 + 7y + 84. What is the greatest common factor of the polynomial?

-7

In general, the complete factorization of a polynomial results from following these steps:

1. Arrange the terms in descending powers of a variable. 2. Factor out the GCF if there is one (or its negative to make the first term positive). 3. If the resulting polynomial is a difference of two squares, use A 2 - B 2 = (A + B)(A - B) to factor it. (Recall that the sum of two squares is prime.) 4. If the resulting polynomial has three terms, use the factoring procedures for trinomials. 5. If the resulting polynomial has four terms, use factoring by grouping. 6. Be sure that all factors are prime. Sometimes the factoring procedure will need to be repeated with any resulting factor that is itself factorable. 7. Check the factorization by multiplying the factors together.

Factor 2x^4 - 20x^2 - 78

2(x^2 + 3)(x^2 - 13)

Which of the following is the complete factorization of 25a^2 - 100

25(a - 2)(a + 2)

Factor 75t^2 + 12

3(25t^2 + 4)

Which of the following is the complete factorization of 3x^2 - 6x - 240

3(x + 8)(x - 10)

Factor 35d + 5d^2 - 30.

5(d^2 + 7d - 6)

Factor 5x^4 - 30x^2 - 135

5(x - 3)(x + 3)(x^2 + 3)

Factor 6y^2 - 48 + 42y

6(y + 8)(y - 1)

Review

When factoring a polynomial, all factors must be prime.

complete factorization

one in which all polynomial factors (excluding monomial factors) are prime; for example, the factorization of 4x² + 8x - 60 to 4(x² + 2x - 15) is not complete because the trinomial x² + 2x - 15 can itself be factored