Question

Find all zeros of ƒ(x) = x^3 + 2x^2 – x – 2. Then determine the multiplicity at each zero. State whether the graph will touch or cross the x-axis at the zero.

Answer 1

Answer:

Zeros: -2, 1, -1

Multiplicity of each zero is 1

The graph crosses the x-axisat each zero.

Step-by-step explanation:

Consider the polynomial function

$f(x) = x^3 + 2x^2 - x- 2.$

First, factor this function:

$f(x) = x^3 + 2x^2 - x- 2=(x^3+2x^2)-(x+2)=x^2(x+2)-(x+2)=(x+2)(x^2-1)=(x+2)(x-1)(x+1).$

The zero of the function is such value of x at which f(x)=0. Note that f(x)=0 when

$(x+2)(x-1)(x+1)=0$

This is possible when

$x=-2\ \text{ or }x=1\text{ or }x=-1.$

Therefore, the function f(x) has three zeros. Each zero is of multiplicity 1 (because each factor has 1st degree)

This means that at each zero the function will crosss the x-axis. (The function will touch the x-axis at its zero when this zero has even multiplicity)