Question

Determine if the function is a polynomial function. If the function is a polynomial function, state the degree and the leading coefficient. If the function is not a polynomial, state why. f(x)=(2x+3)^2(3x+5)^2

This is not a polynomial function because there is no leading coefficient.



This is a polynomial function of degree 4 with a leading coefficient of 36.



This is a polynomial function of degree 4 with a leading coefficient of −36.



This is not a polynomial function as the factors are not all linear.

Answer 1

The correct option is 2. The given function is a polynomial of degree 4 with a leading coefficient of 36.

Explanation:

A polynomial function is defined as

`P=a_0x^n+a_1x^(n-1)+....+a_(n-1)x+a_n`

The given function is

`f(x)=(2x+3)^2(3x+5)^2`

Using perfect square trinomial property.

`f(x)=(4x^2+12x+9)(9x^2+30x+25)`
`[\because (a+b)^2=a^2+2ab+b^2]`

Using distributive property, we get

`f(x)=4x^2(9x^2+30x+25)+12x(9x^2+30x+25)+9(9x^2+30x+25)`

`f(x)=36x^4+120x^3+100x^2+108x^3+360x^2+300x+81x^2+270x+225`

`f(x)=36x^4+228x^3+541x^2+570x+225`

The given function is a polynomial of degree 4 with a leading coefficient of 36.

Therefore the correct option is 2.