# Lesson 12

Polynomial Division (Part 1)

### Problem 1

The polynomial function $$p(x)=x^3-3x^2-10x+24$$ 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.

### Problem 2

Tyler thinks he knows one of the linear factors of $$P(x)=x^3-9x^2+23x-15$$. After finding that $$P(1)=0$$, he suspects that $$x-1$$ is a factor of $$P(x)$$. Here is the diagram he made to check if he’s right, but he set it up incorrectly. What went wrong?

$$x^2$$ $$\text-8x$$ -15
$$x$$ $$x^3$$ $$\text-8x^2$$ $$\text-15x$$
1 $$x^2$$ $$\text-8x$$ -15

### Problem 3

The polynomial function $$q(x)=2x^4-9x^3-12x^2+29x+30$$ has known factors $$(x-2)$$ and $$(x+1)$$. Which expression represents $$q(x)$$ as the product of linear factors?

A:

$$(2x - 5)(x+3)(x-2)(x+1)$$

B:

$$(2x + 3)(x-5)(x-2)(x+1)$$

C:

$$(2x + 15)(x-1)(x-2)(x+1)$$

D:

$$(2x - 15)(x+1)(x-2)(x+1)$$

### Problem 4

Each year a certain amount of money is deposited in an account which pays an annual interest rate of $$r$$ so that at the end of each year the balance in the account is multiplied by a growth factor of $$x=1+r$$. $1,000 is deposited at the start of the first year, an additional$300 is deposited at the start of the next year, and \$500 at the start of the following year.

1. Write an expression for the value of the account at the end of three years in terms of the growth factor $$x$$.
2. Determine (to the nearest cent) the amount in the account at the end of three years if the interest rate is 4%.

### Solution

(From Unit 2, Lesson 2.)

### Problem 5

State the degree and end behavior of $$f(x)=5 + 7x - 9x^2 + 4x^3$$. Explain or show your reasoning.

### Solution

(From Unit 2, Lesson 8.)

### Problem 6

Describe the end behavior of $$f(x) = 1 + 7x + 9x^3 + 6x^4 - 2x^5$$.

### Solution

What are the points of intersection between the graphs of the functions $$f(x)=(x+3)(x-1)$$ and $$g(x)=(x+1)(x-3)$$?