practice & problem solving
use long division to divide.
24. ( x^4 + 2x^3 - 8x^2 - 3x + 1 ) divided by ( x + 2 )
use synthetic division to divide.
25. ( x^4 + 5x^3 + 7x^2 - 2x + 17 ) divided by ( x - 3 )
26. make sense and persevere a student divided ( f(x) = x^3 + 8x^2 - 9x - 3 ) by ( x - 2 ) and got a remainder of 19. explain how the student could verify the remainder is correct.
27. reason the area of a rectangle is ( 4x^3 + 14x^2 - 18 ) in.(^2). the length of the rectangle is ( x + 3 ) in. what is the width of the rectangle?

Question

Accepted Answer

### Problem 24: Long Division of Polynomials
# Explanation:
## Step 1: Set up long division
Divide $ x^4 + 2x^3 - 8x^2 - 3x + 1 $ by $ x + 2 $. The dividend is $ x^4 + 2x^3 - 8x^2 - 3x + 1 $ and the divisor is $ x + 2 $.
## Step 2: Divide the leading terms
Divide $ x^4 $ by $ x $ to get $ x^3 $. Multiply $ x + 2 $ by $ x^3 $ to get $ x^4 + 2x^3 $. Subtract this from the dividend:
$$
\begin{align*}
&(x^4 + 2x^3 - 8x^2 - 3x + 1) - (x^4 + 2x^3)\
=& -8x^2 - 3x + 1
\end{align*}
$$
## Step 3: Repeat the process
Divide $ -8x^2 $ by $ x $ to get $ -8x $. Multiply $ x + 2 $ by $ -8x $ to get $ -8x^2 - 16x $. Subtract:
$$
\begin{align*}
&(-8x^2 - 3x + 1) - (-8x^2 - 16x)\
=& 13x + 1
\end{align*}
$$
## Step 4: Continue
Divide $ 13x $ by $ x $ to get $ 13 $. Multiply $ x + 2 $ by $ 13 $ to get $ 13x + 26 $. Subtract:
$$
\begin{align*}
&(13x + 1) - (13x + 26)\
=& -25
\end{align*}
$$
# Answer:
The quotient is $ x^3 - 8x + 13 $ and the remainder is $ -25 $, so $ \frac{x^4 + 2x^3 - 8x^2 - 3x + 1}{x + 2} = x^3 - 8x + 13 - \frac{25}{x + 2} $

### Problem 25: Synthetic Division of Polynomials
# Explanation:
## Step 1: Set up synthetic division
For dividing $ x^4 + 5x^3 + 7x^2 - 2x + 17 $ by $ x - 3 $, we use the root $ r = 3 $. The coefficients of the dividend are $ 1, 5, 7, -2, 17 $.
## Step 2: Perform synthetic division
Bring down the first coefficient (1). Multiply by 3: $ 1 \times 3 = 3 $. Add to the next coefficient: $ 5 + 3 = 8 $.
Multiply 8 by 3: $ 8 \times 3 = 24 $. Add to the next coefficient: $ 7 + 24 = 31 $.
Multiply 31 by 3: $ 31 \times 3 = 93 $. Add to the next coefficient: $ -2 + 93 = 91 $.
Multiply 91 by 3: $ 91 \times 3 = 273 $. Add to the last coefficient: $ 17 + 273 = 290 $.
## Step 3: Write the result
The coefficients of the quotient (a cubic polynomial) are $ 1, 8, 31, 91 $ and the remainder is $ 290 $. So the quotient is $ x^3 + 8x^2 + 31x + 91 $ and the remainder is $ 290 $.
# Answer:
$ \frac{x^4 + 5x^3 + 7x^2 - 2x + 17}{x - 3} = x^3 + 8x^2 + 31x + 91 + \frac{290}{x - 3} $

### Problem 26: Verifying the Remainder
# Brief Explanations:
To verify the remainder when dividing $ f(x) = x^3 + 8x^2 - 9x - 3 $ by $ x - 2 $, we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial $ f(x) $ is divided by $ x - a $, the remainder is $ f(a) $. Here, $ a = 2 $, so we evaluate $ f(2) $.
Calculate $ f(2) = (2)^3 + 8(2)^2 - 9(2) - 3 = 8 + 32 - 18 - 3 = 19 $, which matches the given remainder. Alternatively, we can perform the division (long or synthetic) and check the remainder.
# Answer:
The student can use the Remainder Theorem: evaluate $ f(2) $ for $ f(x) = x^3 + 8x^2 - 9x - 3 $. If $ f(2) = 19 $, the remainder is correct. Calculating $ f(2) = 8 + 32 - 18 - 3 = 19 $, so the remainder is correct.

### Problem 27: Width of a Rectangle
# Explanation:
## Step 1: Recall the area formula
The area of a rectangle is $ A = length \times width $, so $ width = \frac{area}{length} $. Here, $ A = 4x^3 + 14x^2 - 18 $ and $ length = x + 3 $. We need to divide $ 4x^3 + 14x^2 - 18 $ by $ x + 3 $ using polynomial long division.
## Step 2: Divide the leading terms
Divide $ 4x^3 $ by $ x $ to get $ 4x^2 $. Multiply $ x + 3 $ by $ 4x^2 $ to get $ 4x^3 + 12x^2 $. Subtract from the dividend:
$$
\begin{align*}
&(4x^3 + 14x^2 - 18) - (4x^3 + 12x^2)\
=& 2x^2 - 18
\end{align*}
$$
## Step 3: Repeat
Divide $ 2x^2 $ by $ x $ to get $ 2x $. Multiply $ x + 3 $ by $ 2x $ to get $ 2x^2 + 6x $. Subtract:
$$
\begin{align*}
&(2x^2 - 18) - (2x^2 + 6x)\
=& -6x - 18
\end{align*}
$$
## Step 4: Continue
Divide $ -6x $ by $ x $ to get $ -6 $. Multiply $ x + 3 $ by $ -6 $ to get $ -6x - 18 $. Subtract:
$$
\begin{align*}
&(-6x - 18) - (-6x - 18)\
=& 0
\end{align*}
$$
# Answer:
The width is $ 4x^2 + 2x - 6 $ inches.

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QUESTION IMAGE

Question

Explanation:

Problem 24: Long Division of Polynomials

Step 1: Set up long division

Step 2: Divide the leading terms

Step 3: Repeat the process

Step 4: Continue

Step 1: Set up synthetic division

Step 2: Perform synthetic division

Step 3: Write the result

Answer:

Problem 25: Synthetic Division of Polynomials