Ch 10: Polynomials E) Factoring x 2 + bx + c Objective: To factor polynomials when a = 1

Definitions Polynomial in Standard Form: ax 2 + bx + c = 0 A polynomial written in descending order based on the exponents. Polynomial in Factored Form: (x + c 1 )( x + c 2 ) = 0 A polynomial written as the product of two (or more) binomials.

Table Method 1)Arrange the polynomial in Standard Form everything on one side and zero on the other 2)Create a table (see below) listing all of the factors of c in columns 1 and 2 3)Place the sum of columns 1 & 2 into column 3 4)Look for the value of b in column 3 and place the values from columns 1 & 2 into the Factored Form. (x + )( x + ) = 0 x 2 + 7x + 12 = 0 1  12, 2  6, 3  4, and the negatives −1−1 − 12 −2−2 −6−6 −3−3 −4−4 = − 13 = − 8 = − 7 = 13 = 8 = = b c1c1 c2c2 c1c1 c2c2

Table Method Example 1 Example 2 x 2 – 8x + 12 = 0x 2 + x − 12 = 0 (x + )( x + ) = −1−1 − 12 −2−2 −6−6 −3−3 −4−4 = − 13 = − 8 = − 7 = 13 = 8 = = b c1c1 c2c2 c1c1 c2c2 (x + )( x + ) = 0 1 − 12 2 −6−6 3 −4−4 −1−112 −2−26 −3−34 = 11 = 4 = 1 = − 11 = − 4 = − = b c1c1 c2c2 c1c1 c2c2 (x − 2)( x − 6) = 0(x − 3)( x + 4) = 0

“X” Method 1)Arrange the polynomial in Standard Form everything on one side and zero on the other 2)Place the value of b at the top of the “x” 3)Place the value of c at the bottom of the “x” 4)Find the values that can be inserted into the left & right side of the “x” whose sum is on the top and product is on the bottom. use trial & error 5)Place those values into the Factored Form x 2 + 7x + 12 = ≠ 7 add (+) multiply (  ) ≠ = 7 (x + )( x + ) = 0 c1c1 c2c2

“X” Method Example 1 Example 2 −4−4 −5−5 1 −5−5 (x + )( x + ) = 0 c1c1 c2c2 x 2 – 6x + 9 = 0x 2 – 4x − 5 = 0 add (+) multiply (  ) −6−6 9 −3−3 −3−3 (x + )( x + ) = 0 c1c1 c2c2 add (+) multiply (  ) (x + 1)( x − 5) = 0(x − 3)( x − 3) = 0

Algebra Tiles 1)Arrange the polynomial in Standard Form everything on one side and zero on the other 2)Lay out the tiles that represent the polynomial x 2 + 7x + 12 = 0 7x12 x2x2 3) Rearrange the tiles so they form a rectangle 4) The tiles across the bottom and down the right side represent the binomials in Factored Form.

Algebra Tiles 1)Arrange the polynomial in Standard Form everything on one side and zero on the other 2)Lay out the tiles that represent the polynomial x 2 + 7x + 12 = 0 7x12 x2x2 3) Rearrange the tiles so they form a rectangle 4) The tiles across the bottom and down the right side represent the binomials in Factored Form. x + 4 x = (x + 4)(x + 3)

Algebra Tiles Example 1 Example 2 x 2 − 3x − 4 = 0x 2 + 5x + 6 = x + 2 x + 3 (x + 2)(x + 3)= x − 4 - x − 1 (x − 4)(-x − 1)= 0 (x − 4)( x + 1) = 0

Classwork 1) 3) 2) 4) x 2 − 8x + 15 = 0x 2 − 16x + 63 = 0 x x + 48 = 0 x 2 + 2x − 15 = 0 (x − 5)(x − 3) = 0(x − 7)(x − 9) = 0 (x + 6)(x + 8) = 0(x + 5)(x − 3) = 0

5) 7) 6) 8) x 2 − 2x + 1 = 0x2x2 x 2 + 2x − 8 = 0x x + 30 = 0 (x − 1)(x − 1) = 0(x − 1)(x + 1) = 0 (x − 2)(x + 4) = 0 (x + 5)(x + 6) = 0 − 1 = 0 + 0x