Creating Polynomials Given the Zeros.. What do we already know about polynomial functions? They are either ODD functions They are either EVEN functions.

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Presentation transcript:

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 Fifth Power y = 4x 5 –x + 5 Quadratics y = 4x Quartics y = 4x Quadratics y = 4x 2 - 5

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

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

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.

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. x 3 - 5x 2 +2x + 8 Step 3: Name it! f(x) =x 3 - 5x 2 +2x + 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.

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.

x 2 i x 2 -i x2x2 2x ix 2i 4 2x -ix-2i -i 2 1 x x x x (x 2 + 4x + 5) x -2 x -2 x2x2 -2x 4 -3 x x x x (x 2 - 4x + 1)

x 2 -4x 1 x 2 + 4x + 5 x4x4 4x 3 -3 f(x) = x 4 -10x 2 -16x x 3 x2x2 -16x 2 5x x 54x