4. Polynomial Functions

Understanding Polynomial Functions

4. Polynomial Functions

Understanding Polynomial Functions

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Multiple Choice

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=2x^4-12x^3+18x^2$

Touch at $x=0$ , Cross at $x=-3$

Touch at $x=0$ , Touch at $x=3$

Touch at $x=1,$ Cross at $x=-3$

Touch at $x=-1$ , Cross at $x=0$

Verified step by step guidance

First, factor out the greatest common factor from the polynomial f(x) = 2x^4 - 12x^3 + 18x^2. Notice that each term has a factor of 2x^2.

After factoring out 2x^2, the polynomial becomes f(x) = 2x^2(x^2 - 6x + 9).

Next, factor the quadratic expression x^2 - 6x + 9. Recognize that this is a perfect square trinomial, which can be factored as (x - 3)^2.

Now, the polynomial is expressed as f(x) = 2x^2(x - 3)^2. Identify the zeros by setting each factor equal to zero: 2x^2 = 0 and (x - 3)^2 = 0.

Solve for x in each equation: 2x^2 = 0 gives x = 0 with multiplicity 2, and (x - 3)^2 = 0 gives x = 3 with multiplicity 2. Since both zeros have even multiplicities, the graph touches the x-axis at x = 0 and x = 3.

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Struggling with College Algebra?

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. f(x)=2x4−12x3+18x2f\left(x\right)=2x^4-12x^3+18x^2f(x)=2x4−12x3+18x2

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Understanding Polynomial Functions practice set

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=2x^4-12x^3+18x^2$