Lesson 14

What Do You Know About Polynomials?

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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.

Blank coordinate plane, x, negative 10 to 10 by twos, y axis unlabeled.

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?

 

Coordinate plane, x, negative 10 to 10 by 1, y axis unlabeled. Curve begins in the third quadrant, through negative 1 comma 0, has a positive y-intercept, through 2 comma 0 and 6 comma 0.

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.

Coordinate plane, x, negative 10 to 10 by 1, y axis unlabeled. Curve begins in the second quadrant, through negative 4 comma 0, negative 1 comma 0, through 2 comma 0 and 6 comma 0.
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.

graph of rational function. x intercepts of -2 and 1. graph changes direction, from decreasing to increasing at x=1.

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.)