Lesson 13

Polynomial Division (Part 2)

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Problem 1

The polynomial function \(B(x)=x^3-21x+20\) has a known factor of \((x-4)\). Rewrite \(B(x)\) as a product of linear factors.

Solution

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Problem 2

Let the function \(P\) be defined by \(P(x) = x^3 + 7x^2 - 26x - 72\) where \((x+9)\) is a factor. To rewrite the function as the product of two factors, long division was used but an error was made:

\(\displaystyle \require{enclose} \begin{array}{r}  x^2+16x+118\phantom{000} \\ x+9 \enclose{longdiv}{x^3+7x^2-26x-72} \phantom{000}\\    \underline{-x^3+9x^2} \phantom{-26x-720000} \\  16x^2-26x \phantom{-720000}\\ \underline{-16x^2+144x} \phantom{-20000} \\ 118x-72 \phantom{00} \\ \underline{-118x+1062} \\ 990 \end{array}\)

How can we tell by looking at the remainder that an error was made somewhere?

Solution

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Problem 3

For the polynomial function \(A(x)=x^4-2x^3-21x^2+22x+40\) we know \((x-5)\) is a factor. Select all the other linear factors of \(A(x)\).

A:

\((x+1)\)

B:

\((x-1)\)

C:

\((x+2)\)

D:

\((x-2)\)

E:

\((x+4)\)

F:

\((x-4)\)

G:

\((x+8)\)

Solution

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Problem 4

Match the polynomial function with its constant term.

A:

\(P(x)=(x-2)(x-3)(x+7)\)

B:

\(P(x)=(x+2)(x-3)(x+7)\)

C:

\(P(x)=\frac12(x-2)(x-3)(x+7)\)

D:

\(P(x)=5(x-2)(x-3)(x+7)\)

E:

\(P(x)=\text-5(x-2)(x-3)(x+7)\)

1:

-210

2:

-42

3:

21

4:

42

5:

210

Solution

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(From Unit 2, Lesson 6.)

Problem 5

What are the solutions to the equation \((x-2)(x-4)=8\)?

Solution

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(From Unit 2, Lesson 11.)

Problem 6

The graph of a polynomial function \(f\) is shown. Which statement is true about the end behavior of the polynomial function?

polynomial function graphed. x intercepts = -4, 0, and 3. y intercept = 0. f of x increases as x increases. 
A:

As \(x\) gets larger and larger in the either the positive or the negative direction, \(f(x)\) gets larger and larger in the positive direction.

B:

As \(x\) gets larger and larger in the positive direction, \(f(x)\) gets larger and larger in the positive direction. As \(x\) gets larger and larger in the negative direction, \(f(x)\) gets larger and larger in the negative direction.

C:

As \(x\) gets larger and larger in the positive direction, \(f(x)\) gets larger and larger in the negative direction. As \(x\) gets larger and larger in the negative direction, \(f(x)\) gets larger and larger in the positive direction.

D:

As \(x\) gets larger and larger in the either the positive or negative direction, \(f(x)\) gets larger and larger in the negative direction.

Solution

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(From Unit 2, Lesson 8.)

Problem 7

The polynomial function \(p(x)=x^3+3x^2-6x-8\) has a known factor of \((x+4)\).

  1. Rewrite \(p(x)\) as the product of linear factors.
  2. Draw a rough sketch of the graph of the function.

Solution

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(From Unit 2, Lesson 12.)