4. Polynomial Functions

Understanding Polynomial Functions

4. Polynomial Functions

Understanding Polynomial Functions

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Scelta multipla

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 $x=0$

Guida verificata passo dopo passo

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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Scelta multipla

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f$\left$(x$\right$)=2+x$

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f$\left$(x$\right$)=3x^2+5x+2$

Determine the end behavior of the given polynomial function. $f$\left$(x$\right$)=x^2+4x+x+7x^3$

Match the given polynomial function to its graph based on end behavior. $f$\left$(x$\right$)=-2x^3+x^2+1$

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$)=x^2$\left$(x-1$\right$)^3$\left$(2x+6$\right$)$

Determine the maximum number of turning points for the given polynomial function. $f$\left$(x$\right$)=6x^4+2x$

Based ONLY on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? $f$\left$(x$\right$)=x^3+1$