Lesson 14

What Do You Know About Polynomials?

Problem 1

We know these things about a polynomial function, $$f(x)$$: it has exactly one relative maximum and one relative minimum, it has exactly three zeros, and it has a known factor of $$(x-4)$$. Sketch a graph of $$f(x)$$ given this information.

Solution

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Problem 2

Mai graphs a polynomial function, $$f(x)$$, that has three linear factors $$(x+6)$$, $$(x+2)$$, and $$(x-1)$$. But she makes a mistake. What is her mistake?

Solution

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Problem 3

Here is the graph of a polynomial function with degree 4.

Select all of the statements that are true about the function.

A:

The leading coefficient is positive.

B:

The constant term is negative.

C:

It has 2 relative maximums.

D:

It has 4 linear factors.

E:

One of the factors is $$(x-1)$$.

F:

One of the zeros is $$x=2$$.

G:

There is a relative minimum between $$x=1$$ and $$x=3$$.

Solution

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Problem 4

State the degree and end behavior of $$f(x)=2x^3-3x^5-x^2+1$$. Explain or show your reasoning.

Solution

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(From Unit 2, Lesson 9.)

Problem 5

Is this the graph of $$g(x)=(x-1)^2(x+2)$$ or $$h(x)=(x-1)(x+2)^2$$? Explain how you know.

Solution

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(From Unit 2, Lesson 10.)

Problem 6

Kiran thinks he knows one of the linear factors of $$P(x)=x^3 + x^2 - 17 x + 15$$. After finding that $$P(3)=0$$, Kiran suspects that $$x-3$$ is a factor of $$P(x)$$, so he sets up a diagram to check. Here is the diagram he made to check his reasoning, but he set it up incorrectly. What went wrong?

$$x^2$$ $$4x$$ -5
$$x$$ $$x^3$$ $$4x^2$$ $$\text-5x$$
3 $$3x^2$$ $$12x$$ 15

Solution

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(From Unit 2, Lesson 12.)

Problem 7

The polynomial function $$B(x)=x^3+8x^2+5x-14$$ has a known factor of $$(x+2)$$. Rewrite $$B(x)$$ as a product of linear factors.

Solution

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(From Unit 2, Lesson 13.)