1. Factoring Polynomials to Find Roots

2. Factoring a Polynomial to Find Roots Factor 3x 4 + 16x 3 + 21x 2 + 14x – 24 and find all of the roots. Roots: Factors: Factor the Polynomial: -4 (x+4) The # inside of the factor must divide the constant term in the Standard Form (-24÷4=-6) y = 3x 4 + 16x 3 + 21x 2 + 14x – 24 Use the graph to find a root Use polynomial division to find the missing factor

3. Factoring a Polynomial to Find Roots Goal: Factor 3x 4 + 16x 3 + 21x 2 + 14x – 24 (by x + 4) 3x 4 +16x 3 +21x 2 +14x – 24 x + 4 3x33x3 3x43x4 12x 3 4x34x3 4x24x2 16x 2 5x25x2 5x5x 20x -6x -6 -24 (x + 4)(3x 3 + 4x 2 + 5x – 6)

4. Factoring a Polynomial to Find Roots Factor 3x 4 + 16x 3 + 21x 2 + 14x – 24 and find all of the roots. Roots: Factors: -4 (x+4), 2/3, (3x-2) When does this equal 0? y = 3x 4 + 16x 3 + 21x 2 + 14x – 24 y = 3x 3 + 4x 2 + 5x – 6 Work backwards to find factor: x=2/3 3x=2 3x – 2=0 Factor the Polynomial: Use the graph to find another root Use polynomial division again

5. Factoring a Polynomial to Find Roots Goal: Factor 3x 3 + 4x 2 + 5x – 6 (by 3x – 2) 3x 3 + 4x 2 +5x –6 3x -2 x2x2 3x33x3 -2x 2 6x26x2 2x2x -4x 9x9x 3 -6

6. Factoring a Polynomial to Find Roots Roots: Factors: -4 (x+4), 2/3, (3x-2) Factor 3x 4 + 16x 3 + 21x 2 + 14x – 24 and find all of the roots. Factor the Polynomial: y = 3x 3 + 4x 2 + 5x – 6 y = x 2 + 2x + 3 If a factor has no real roots, it can not be factored more. Do NOT include irrational or complex numbers in the factored version. Now use the quadratic formula to find the complex roots. When does this equal 0? x 2 +2x+3,

7. Factoring a Polynomial to Find Roots Find the roots of: x 2 + 2x + 3 and Remember: Complex Roots will always come in pairs.

8. Summary Roots:-4, 2/3, Factor 3x 4 + 16x 3 + 21x 2 + 14x – 24 and find all of the roots. y = 3x 4 + 16x 3 + 21x 2 + 14x – 24 Factor the Polynomial: