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.

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erinna erinna · Answer 1 · 2019-07-10T14:43:11+02:00

Answer:

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

Step-by-step explanation:

A polynomial function is defined as

[tex]P=a_0x^n+a_1x^{n-1}+....+a_{n-1}x+a_n[/tex]

The given function is

[tex]f(x)=(2x+3)^2(3x+5)^2[/tex]

Using perfect square trinomial property.

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

Using distributive property, we get

[tex]f(x)=4x^2(9x^2+30x+25)+12x(9x^2+30x+25)+9(9x^2+30x+25)[/tex]

[tex]f(x)=36x^4+120x^3+100x^2+108x^3+360x^2+300x+81x^2+270x+225[/tex]

[tex]f(x)=36x^4+228x^3+541x^2+570x+225[/tex]

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

Therefore the correct option is 2.