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Factoring a Polynomial Example 1: Factoring a Polynomial Completely factor x3 + 2x2 – 11x – 12 Use the graph or table to find at least one real root. x = -4 is a real root because it is an x-intercept. Since x = -4 is a root, (x + 4) is a factor of the original cubic equation. Now use polynomial division to “factor out” the (x + 4). y x 2 x 11 x 12 3 2 Example 1: Factoring a Polynomial Completely factor x3 + 2x2 – 11x – 12 x2 3 -2x 2 -3 x x -2x -3x +4 4x2 -8x -12 x3 + 2x2 – 11x – 12 Thus, the completely factored form is: Now we can rewrite the cubic: x 4 x 2 2 x 3 This quadratic can be factored using old techniques: (x + 1)(x – 3) Since the graph of the cubic had more than one real root, this may be able to be factored more. x 4 x 1 x 3 Let’s try another example. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Use the graph or table to find at least one real root. x = -2 is a real root because it is an x-intercept. Since x = -2 is a root, (x + 2) is a factor of the original degree 4 equation. Now use polynomial division to “factor out” the (x + 2). y x x 4 x 16 4 3 Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 x3 -3x2 6x -8 Now we can rewrite the degree 4 equation: x 2 x 3x 6 x 8 3 x x 4 -3x 3 2 6x + 2 2x3 -6x2 12x -8x -16 x4 – x3 + 0x2 + 4x – 16 Make sure to include all powers of x 2 Let’s check the graph of this cubic to see if it has a real root. Since the graph of the degree 4 equation had more than one real root, this may be able to be factored more. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Current Factored form: x 2 x 3 x 6 x 8 Use the graph or table of the cubic in the factored form to find at least one real root. 3 2 x = 2 is a real root because it is an x-intercept. Since x = 2 is a root, (x – 2) is a factor of the cubic in the factored form. y x 3x 6 x 8 3 2 Now use polynomial division to “factor out” the (x – 2) of the cubic in the factored form. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Current Factored form: x 2 x 3 x 6 x 8 Now we can rewrite the 2 x -x 4 current factored form as: 3 x x3 -x2 4x –2 -2x2 2x -8 x3 – 3x2 + 6x Thus, the completely factored form is: –8 2 x 2 x 2 x 2 x 4 This quadratic can NOT be factored using old techniques (No x-intercepts). Since the graph of the cubic had only one real root, this may NOT be able to be factored more. x 2 x 2 x 2 x 4