The polynomial of degree 3 , P ( x ) , has a root of multiplicity 2 at x = 4 and a root of multiplicity 1at x = − 4 . The y -intercept is y = − 38.4 . Find a formula for P ( x ) .

Question

The polynomial of degree 33, P(x)P(x), has a root of multiplicity 22 at x=4x=4 and a root of multiplicity 11 at x=−4x=-4. The yy-intercept is y=−38.4y=-38.4.

Find a formula for P(x)P(x).

Answer 1

If 4 is a root, then (x-4) is a factor of the polynomial function. Since the multiplicity is 2: (x-4)² is a factor.

Since -4 is a root, (x+4) is a factor with multiplicity 1.

So far we have P(x) = A(x-4)²(x+4) where A is nonzero.

To find A, use the fact that the point (0,-38.4) lies on the graph, i.e. y-intercept has x-coordinate 0.

-38.4 = A(-4)²(4)

A = -38.4/64 = -3/5

So P(x) = -3/5 (x³-4x²-16x+64). You can distribute the -3/5.

Note that (x-4)(x-4)(x+4) = (x-4)(x²-16) = now use FOIL.

Here is a graph depicting this polynomial function:

desmos.com/calculator/t8ihmoy5de

Use the slider on A and note the roots remain the same. Also where multiplicity is even, the graph bounces off the x-axis. Where multiplicity is odd, the graph passes through the x-axis.

Answer 2

P(x) = a(x - 4)²(x + 4)

-38.4 = a(0 - 4)²(0 + 4)

Can you sovle for a and answer?