Question

complete each statement so it means the same as \4 is a zero of the function.\ the graph of the function crosses the ____ at 4. ____ is a factor of the polynomial. the graph of the function crosses the at 4. is a factor of the polynomial.

Explanation:

Step1: Recall the definition of a zero of a function

A zero of a function \( f(x) \) is a value \( x = a \) such that \( f(a)=0 \). Geometrically, this means the graph of the function intersects the \( x \)-axis (since \( y = 0 \) on the \( x \)-axis) at \( x=a \). So for \( a = 4 \), the graph crosses the \( x \)-axis at 4.

Step2: Recall the Factor Theorem

The Factor Theorem states that if \( r \) is a zero of a polynomial \( f(x) \), then \( (x - r) \) is a factor of \( f(x) \). Since 4 is a zero of the function, by the Factor Theorem, \( (x - 4) \) is a factor of the polynomial.

Answer:

The graph of the function crosses the \( x \)-axis at 4. \( (x - 4) \) is a factor of the polynomial.