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Creating Polynomials Given the Zeros.

Creating Polynomials Given the Zeros. What do we already know about polynomial functions?. They are either ODD functions. They are either EVEN functions. Linear y = 4x - 5. Cubic y = 4x 3 - 5. Quadratics y = 4x 2 - 5. Quadratics y = 4x 2 - 5. Fifth Power y = 4x 5 –x + 5.

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Creating Polynomials Given the Zeros.

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  1. Creating Polynomials Given the Zeros.

  2. What do we already know about polynomial functions? They are either ODD functions They are either EVEN functions Linear y = 4x - 5 Cubic y = 4x3 - 5 Quadratics y = 4x2 - 5 Quadratics y = 4x2 - 5 Fifth Power y = 4x5 –x + 5 Quartics y = 4x4 - 5

  3. We know that factoring and then solving those factors set equal to zero allows us to find possible x intercepts. TOOLS WE’VE USED Long Division (works on all factors of any degree) Factoring GCF Quadratic Formula (x + )(x + ) Synthetic Division (works only with factors of degree 1) The “6” step Grouping Cubic** p/q

  4. We know that solutions of polynomial functions can be rational, irrational or imaginary. X intercepts are real. Zeros are x-intercepts if they are real Zeros are solutions that let the polynomial equal 0

  5. We have seen that imaginaries and square roots come in pairs ( + or -). So we could CREATE a polynomial if we were given the polynomial’s zeros.

  6. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros. -1, 2, 4 Step 1: Turn the zeros into factors. (x+1)(x- 2)(x- 4) Step 2: Multiply the factors together. f(x) =x3 - 5x2 +2x + 8 x3 - 5x2 +2x + 8 Step 3: Name it!

  7. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros. Step 1: Turn the zeros into factors. Must remember that “i”s and roots come in pairs. Step 2: Multiply factors.

  8. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros. Step 1: Turn the zeros into factors. Must remember that “i”s and roots come in pairs. Step 2: Multiply factors.

  9. x 2 i x -2 x x x 2 -i x2 -2x x2 ix 2x x -2 x 4 x 2i 2x -2x 4 x x x x -ix -2i 1 -i2 -3 (x2- 4x + 1) (x2+ 4x + 5)

  10. x2+ 4x + 5 x2 -4x 1 4x3 x4 5x2 -20x -4x3 -16x2 4x 5 x2 f(x) = x4-10x2 -16x + 5 -3

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