Algebra 1 Section 10.2

product of the constants Factoring x2 + bx + c (x + 5)(x + 7) = x2 + 12x + 35 constants sum of the constants product of the constants

Factoring x2 + bx + c Write the factors of the first term. Find factors of the constant term, c, whose sum is b. Write these factors as the last terms in the binomials.

Factoring x2 + bx + c Check the results by multiplying.

Example 1 Factor x2 – 5x + 6. 3. Write these factors as the last terms in the binomials. 4. Check by multiplying. Find two factors of +6 whose sum is -5. 1. Write the factors of x2 as the first terms in the binomials. -2 and -3 (x – 2)(x – 3) (x )(x ) x2 – 2x – 3x + 6 x2 – 5x + 6

Find two factors of -6 whose sum is +5. Example 2 Factor y2 + 5y – 6. Find two factors of -6 whose sum is +5. 6 and -1 (y + 6)(y – 1) (y )(y )

Determining the Signs of the Factors of c If the sign of c is positive, a. the signs are the same and b. the factors will have the sign of the trinomial’s middle term.

Determining the Signs of the Factors of c If the sign of c is negative, a. the signs are different and b. the larger factor will have the sign of the trinomial’s middle term.

Find two factors of -72 whose sum is +1. Example 3 Factor x2 + x – 72. Find two factors of -72 whose sum is +1. 9 and -8 (x + 9)(x – 8) (x )(x )

Example 4 Factor x2 + 3x + 6. Find two factors of 6 whose sum is +3. There are no such factors. Therefore, this trinomial is prime. (x )(x )

Factoring The first step in factoring a polynomial is to factor out any common monomial factors. When the constant term is a large number, a more organized approach may be helpful.

Example 5 Factor 2a2 – 64a – 288. First, factor out the common factor of 2. 2(a2 – 32a – 144) Then, find two factors of -144 that have a sum of -32.

Example 5 2(a2 – 32a – 144) Two factors of -144 that have a sum of -32 are +4 and -36. 2(a + 4)(a – 36) It is still wise to check your answer by multiplying!

Factoring Some trinomials contain two variables. When factoring a trinomial of the form x2 + bxy + cy2, be sure to include a factor of y in the second term of each binomial.

Find two factors of +24 whose sum is +11. Example 6 Factor x2 + 11xy + 24y2. Find two factors of +24 whose sum is +11. 3 and 8 (x + 3y)(x + 8y) (x y)(x y)

Homework: pp. 411-413