Factoring Polynomials

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Keep in mind that, because some of our pairs are not identical factors, we may have to exchange some pairs of factors and make 2 attempts before we can definitely decide a particular pair of factors will not work.

Example

Continued.

Slide 24

We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to 20x.

{x, 25x} {1, 4} (x + 1)(25x + 4) 4x 25x 29x

(x + 4)(25x + 1) x 100x 101x

{x, 25x} {2, 2} (x + 2)(25x + 2) 2x 50x 52x

Factoring Polynomials

Example Continued

Continued.

Slide 25

Check the resulting factorization using the FOIL method.

(5x + 2)(5x + 2) =

= 25x2 + 10x + 10x + 4

= 25x2 + 20x + 4

So our final answer when asked to factor 25x2 + 20x + 4 will be (5x + 2)(5x + 2) or (5x + 2)2.

Factoring Polynomials

Example Continued

Slide 26

Factor the polynomial 21x2 – 41x + 10.

Possible factors of 21x2 are {x, 21x} or {3x, 7x}.

Since the middle term is negative, possible factors of 10 must both be negative: {-1, -10} or {-2, -5}.

We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors.

Factoring Polynomials

Example

Continued.

Slide 27

We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to 41x.

{x, 21x}{1, 10}(x – 1)(21x – 10) –10x 21x – 31x

(x – 10)(21x – 1) –x 210x – 211x

{x, 21x} {2, 5} (x – 2)(21x – 5) –5x 42x – 47x

(x – 5)(21x – 2) –2x 105x – 107x

Factoring Polynomials

Example Continued

Continued.

Slide 28

{3x, 7x}{1, 10}(3x – 1)(7x – 10) 30x 7x 37x

(3x – 10)(7x – 1) 3x 70x 73x

{3x, 7x} {2, 5} (3x – 2)(7x – 5) 15x 14x 29x

Factoring Polynomials

Example Continued

Continued.

Slide 29

Check the resulting factorization using the FOIL method.

(3x – 5)(7x – 2) =

= 21x2 – 6x – 35x + 10

= 21x2 – 41x + 10

So our final answer when asked to factor 21x2 – 41x + 10 will be (3x – 5)(7x – 2).

Factoring Polynomials

Example Continued

Slide 30

Factor the polynomial 3x2 – 7x + 6.

The only possible factors for 3 are 1 and 3, so we know that, if factorable, the polynomial will have to look like (3x )(x ) in factored form, so that the product of the first two terms in the binomials will be 3x2.