Find a polynomial f(x) of degree 3 that has the following zeros.
8 (multiplicity 2), -4

Question

Find a polynomial f(x) of degree 3 that has the following zeros.8 (multiplicity 2), -4

nick1337 · Accepted Answer

To find a polynomial function of degree 3 with the given zeros, we can use the zero-product property and the concept of multiplicities.
The zeros are 8 (with multiplicity 2) and -4.
When a zero has a multiplicity greater than 1, it means that it appears multiple times as a root of the polynomial. In this case, the zero 8 has a multiplicity of 2, which means it is a double root.
To find the polynomial, we can start by writing the linear factors corresponding to each zero. For a zero of multiplicity 2, we need two linear factors.
The linear factor corresponding to the zero 8 is

(x - 8)

. Since the multiplicity is 2, we need to include it twice:

(x - 8) (x - 8)

The linear factor corresponding to the zero -4 is

(x - (- 4))

, which simplifies to

(x + 4)

.
Now, we multiply these factors together to obtain the polynomial:

f (x) = (x - 8) (x - 8) (x + 4)

To simplify further, we can expand the expression using the distributive property:

f (x) = (x^{2} - 16 x + 64) (x + 4)

Next, we can multiply the two binomials using the distributive property:

f (x) = x^{3} + 4 x^{2} - 16 x^{2} - 64 x + 64 x + 256

Combining like terms:

f (x) = x^{3} - 12 x^{2} + 256

Therefore, the polynomial function

f (x)

of degree 3 with the given zeros is:

f (x) = x^{3} - 12 x^{2} + 256

Find a polynomial f(x) of degree 3 that

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