Ch 8.5 (part 2) Factoring ax 2 + bx + c using the Grouping method Objective: To factor polynomials when a ≠ 1.

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Ch 8.5 (part 2) Factoring ax 2 + bx + c using the Grouping method Objective: To factor polynomials when a ≠ 1

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

1)Arrange the polynomial in Standard Form exponents in descending order 2)Place the b term at the top of the “x” 3)Place the a  c term 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 Grouping Form 6)Factor each binomial 7)Group common factors Rules bx ax 2 f1xf1xf2xf2x add (+) multiply (  )

Example 1 2x 2 + 5x + 3 5x 6x 2 2x3x (ax 2 + ) (f 2 + ) f1f1 c add (+) multiply (  ) (2x 2 + 2x) (3x + 3) ( )( ) factor each:2x(x + 1) +3(x + 1) group: 2x+ 3x + 1

Example 2 -9x 14x 2 -2x-7x (ax 2 + ) (f 2 + ) f1f1 c add (+) multiply (  ) (2x 2 − 2x) (−7x + 7) ( )( ) factor each:2x(x − 1) −7(x − 1) group: 2x− 7x − 1 2x 2 − 9x + 7

Example 3 -55x -56x 2 x-56x (ax 2 + ) (f 2 + ) f1f1 c add (+) multiply (  ) (7x 2 + x) (−56x − 8) ( )( ) factor each:x(7x + 1) −8(7x + 1) group: x− 87x + 1 7x 2 − 55x − 8

Example 4 4x -12x 2 -2x6x (ax 2 + ) (f 2 + ) f1f1 c add (+) multiply (  ) (4x 2 − 2x) (6x − 3) ( )( ) factor each:2x(2x − 1) +3(2x − 1) group: 2x+ 32x − 1 4x 2 + 4x – 3

Classwork 1) 3) 2) 4) 2x 2 + 7x + 32x 2 − 19x x 2 − 29x x 2 − 7x − 49 (2x + 1)(x + 3)(2x − 3)(x − 8) (5x − 4)(x − 5)(2x + 7)(x − 7)

5) 7) 6) 8) 4x 2 + 4x + 14x 2 7x 2 − 24x − 166x x − 21 (2x + 1)(2x + 1)(2x − 5)(2x + 5) (7x + 4)(x − 4) (2x + 7)(3x − 3) − x