A polynomial function has a root of –6 with multiplicity 1, a root of –2 with multiplicity 3

Question

A polynomial function has a root of –6 with multiplicity 1, a root of –2 with multiplicity 3, a root of 0 with multiplicity 2, and a root of 4 with multiplicity 3. If the function has a positive leading coefficient and is of odd degree, which statement about the graph is true?

Answer 1

When you set the factors of a polynomial equal to zero, you can find the roots.  This is because the roots are the values of x that make the function equal zero.  Multiplicity is the number of times the roots repeat.

 

Root of -6 with multiplicity 1 is the factor (x + 6).

Root of -2 with multiplicity 3 is the factor (x + 2)³.

Root of 0 with multiplicity 2 is the factor x².

Root of 4 with multiplicity 3 is the factor (x - 4)³.

 

 

The function is

 

f(x) = x²(x + 6)(x + 2)³(x - 4)³

 

f(x) = x*x(x + 6)(x + 2)(x + 2)(x + 2)(x - 4)(x - 4)(x - 4)

 

 

If we expand this polynomial, the leading term of the polynomial will be x⁹.  The leading term was obtained by multiplying out the first terms of each binomial together.  The last term of the polynomial will be -3072.  This was obtained by multiplying out the last terms of each binomial together.