Warmup Solve: 𝑥 4 −27𝑥=0

4-9 Roots and ZEROs Determine the number of solutions for a polynomial equation. Find the solutions of a polynomial equation.

Recall that all real numbers can be written as complex numbers a + 0i. 4𝑥 4 −3 𝑥 3 +5𝑥−6=0 𝑥 3 +2 𝑥 2 +6=0 −2𝑥 5 −3 𝑥 2 +8=0

In the last section we learned how to figure out if a binomial (x – r) was a factor of a polynomial. Now we need to learn how to recognize which roots to try. Your graphing calculator can help with that. Graph the associated polynomial function (get zero on one side, replace zero with y, look at x-intercepts where y is zero) If you can get an exact zero, you can form a factor with that zero.

Repeated Roots: You can recognize an even number of repeated roots from a graph of the polynomial if the curve is only tangent to the x-axis at a value x = r, and doesn’t cross at x = r (bounces off the x-axis). You will need to use them an even number of times to get factors or zeros. Find all the zeros of each function.

p. 363 Find all the zeros of the function.

p. 298 Write a polynomial function of least degree with integral coefficients that have the given zeros.