Question

For the polynomial function ƒ(x) = (2 − x)(x + 1)(x − 3)(x + 5), find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept.

Answer 1

Answer:

x=2 mult. 1, x=-1 mult. 1, x=3 mult. 1, x=-5 mult. 1

Since all have odd multiplicities, all with cross through the x-axis and form a W shape to the graph.

Step-by-step explanation:

The zeros of a polynomial are the x-intercepts of the function. To find them, we factor the polynomial and set each factor equal to 0.

This polynomial is already factored so set each to 0 and solve for x.

$(2-x)=0\\x=2\\\\(x+1)=0\\x=1\\\\(x-3)=0\\x=3\\\\(x+5)=0\\x=-5$

This means x=-5, -1, 2, and 3. Each zero or root has a multiplicity - the number of times the factor occurs. This is also known as the exponent of the factor expression.

(2-x) occurs once since it has exponent 1.

(x+1) occurs once since it has exponent 1.

(x-3) occurs once since it has exponent 1.

(x+5) occurs once since it has exponent 1.

The multiplicity of the root determines the behavior of the graph. Even multiplicities touch the x-axis but do not cross through. Odd multiplicities cross through the x-axis.

x=2 mult. 1, x=-1 mult. 1, x=3 mult. 1, x=-5 mult. 1

Since all have odd multiplicities, all with cross through the x-axis and form a W shape to the graph.