Which statement about the polynomial function g(x) is true?
-If all rational roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1.
-If all roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1.
-If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers.
-If the leading coefficient of g(x) is 1, all roots of g(x) = 0 must be integers.

"These are functions of a single independent variable, in which that variable can appear more than once to any integer power."
The general form of a polynomial function is[tex]f(x)=a_nx^n+...+a_1x+a_0[/tex]

"This theorem is used to determine the rational roots of a polynomial function."
"The theorem states that each rational zero(s) of a polynomial with integer coefficients [tex]f(x)=a_nx^n+...+a_1x+a_0[/tex] is of the form [tex]\frac{p}{q}[/tex] ." where p = p is a factor of the constant [tex]a_0[/tex] and q is a factor of the leading coefficient [tex]a_n[/tex] p and q are relatively prime

For given question,

We have the polynomial function g(x)

Let [tex]g(x)=a_nx^n+...+a_1x+a_0[/tex]

The leading coefficient of a polynomial function is the coefficient of the term with the highest degree.

for the polynomial function g(x),

the leading coefficient = [tex]a_n[/tex]

According to the rational root theorem,

the rational root is [tex]\frac{p}{q}=\frac{factor~of~last~term~a_0}{factor~of~the~leading~term~a_n}[/tex]

If the leading coefficient of g(x) is 1,

⇒ [tex]a_n=1[/tex]

then all rational roots of must be integers.

Therefore, the statement that is true about the polynomial function g(x)

'If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers.'

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