Question

The polynomial of degree 3, P(x), has a root of multiplicity 2 at x=5 and a root of multiplicity 1 at x=−5.
The y-intercept is y=−87.5.
Find a formula for P(x).
P(x)=

Answer 1

Answer:

P(x) = $-0.7(x-5)^{2}(x+5)$

Step-by-step explanation:

Root multiplicity of 2 means the root is repeated 2 times. Root multiplicity of 1 means root is repeated only once.

A polynomial of degree 3 means the largest exponent of x is 3.

Here, the polynomial P(x) has root 5 with multiplicity 2 and -5 with multiplicity 1.

So, P(x) = $a(x-5)^{2}(x+5)$

Where, $a$ is a constant.

Now, it's given that the y intercept is -87.5. Therefore, value of x at y intercept is 0.

We obtain $a$ by plugging in -87.5 for P(x) and 0 for x.

$-87.5=a(0-5)^{2}(0+5)$

$-87.5=a(25\times5)\\ a=-\frac{87.5}{125}=-0.7$

Therefore, the polynomial P(x) is given as:

$P(x) =-0.7(x-5)^{2}(x+5)$