Question

fill in the blanks so that the resulting statement is true.
if ( r ) is a zero of even multiplicity, then the graph touches the ( x )-axis and _____ at ( r ). if ( r ) is a zero of odd multiplicity the graph _____ the ( x )-axis at ( r ).

if ( r ) is a zero of even multiplicity, then the graph touches the ( x )-axis and at ( r ). if ( r ) is a zero of odd multiplicity, then the graph the ( x )-axis at ( r ).

Explanation:

When \(r\) is a zero of even multiplicity for a polynomial function \(y = f(x)\), the factor \((x - r)\) appears an even number of times in the factored form of the polynomial. The graph of the polynomial function \(y=f(x)\) will touch the \(x -\)axis at \(x = r\) and turn around (because the sign of the function does not change on either side of \(x=r\) for an even - multiplicity zero).

When \(r\) is a zero of odd multiplicity for a polynomial function \(y = f(x)\), the factor \((x - r)\) appears an odd number of times in the factored form of the polynomial. The graph of the polynomial function \(y = f(x)\) will cross the \(x -\)axis at \(x=r\) (because the sign of the function changes on either side of \(x = r\) for an odd - multiplicity zero).

Answer:

If \(r\) is a zero of even multiplicity, then the graph touches the \(x\) - axis and \(\text{turns around}\) at \(r\). If \(r\) is a zero of odd multiplicity, the graph \(\text{crosses}\) the \(x\) - axis at \(r\).