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Question

find a polynomial f(x) of degree 3that has the following zeros. 7 (multiplicity 2),-3

leave your answer in factored form

Answer 1

If 7 is a zero of the function, then (x - 7) is a factor, Since its multiplicity is 2, that factor occurs twice, i.e. (x-7)².

Since -3 is a zero, [x-(-3)] is a factor (or (x+3).

So, f(x) looks like this: f(x) = A(x-7)²(x+3) where A is any nonzero real number.

Notice that for the root 7 (even multiplicity) the graph "bounces" off the x-axis.

For the root -3 (odd multiplicity of 1) the graph passes through the x-axis.

desmos.com/calculator/onqrjkefxr

Use the slider on A to convince yourself that the roots remain the same regardless of the value of A.