algebra 2: unit 3a properties and operations of polynomials
3a.5 - 3a.6
18. divide the following: $(6x^4 + 4x^3 + 7x^2 - 10) \div (x + 3)$

19. divide the following with long division: $(x^4 + 5x^3 - 4x^2 + x - 10) \div (x^2 + 2)$

20. given $f(x) = 3x^2 + 5x - 12$ and $g(x) = x - 4$. find $f(x) - g(x)$

21. divide the following: $(7x^4 + 3x^3 + 4x^2 - 13) \div (x + 7)$

22. divide the following with long division: $(x^4 + 9x^3 - 2x^2 + x - 11) \div (x^2 + 3)$

Question

algebra 2: unit 3a properties and operations of polynomials
3a.5 - 3a.6
18. divide the following: $(6x^4 + 4x^3 + 7x^2 - 10) \div (x + 3)$

19. divide the following with long division: $(x^4 + 5x^3 - 4x^2 + x - 10) \div (x^2 + 2)$

20. given $f(x) = 3x^2 + 5x - 12$ and $g(x) = x - 4$. find $f(x) - g(x)$

21. divide the following: $(7x^4 + 3x^3 + 4x^2 - 13) \div (x + 7)$

22. divide the following with long division: $(x^4 + 9x^3 - 2x^2 + x - 11) \div (x^2 + 3)$

Accepted Answer

### Problem 18 (assuming typo, let's take $ (6x^4 + 4x^3 + 7x^2 - 10) \div (x + 3) $)
# Explanation:
## Step 1: Set up polynomial long division
Divide the leading term of the dividend $ 6x^4 $ by the leading term of the divisor $ x $, we get $ 6x^3 $. Multiply the divisor $ (x + 3) $ by $ 6x^3 $: $ 6x^3(x + 3)=6x^4 + 18x^3 $. Subtract this from the dividend:
$ (6x^4 + 4x^3 + 7x^2 - 10)-(6x^4 + 18x^3)= -14x^3 + 7x^2 - 10 $

## Step 2: Continue the division
Divide the leading term $ -14x^3 $ by $ x $, get $ -14x^2 $. Multiply divisor by $ -14x^2 $: $ -14x^2(x + 3)= -14x^3 - 42x^2 $. Subtract:
$ (-14x^3 + 7x^2 - 10)-(-14x^3 - 42x^2)= 49x^2 - 10 $

## Step 3: Next term
Divide $ 49x^2 $ by $ x $, get $ 49x $. Multiply divisor by $ 49x $: $ 49x(x + 3)= 49x^2 + 147x $. Subtract:
$ (49x^2 - 10)-(49x^2 + 147x)= -147x - 10 $

## Step 4: Last term
Divide $ -147x $ by $ x $, get $ -147 $. Multiply divisor by $ -147 $: $ -147(x + 3)= -147x - 441 $. Subtract:
$ (-147x - 10)-(-147x - 441)= 431 $

# Answer: $ 6x^3 - 14x^2 + 49x - 147 + \frac{431}{x + 3} $

### Problem 19: $ (x^4 + 5x^3 - 4x^2 + x - 10) \div (x^2 + 2) $
# Explanation:
## Step 1: Divide leading terms
Divide $ x^4 $ by $ x^2 $ to get $ x^2 $. Multiply divisor $ (x^2 + 2) $ by $ x^2 $: $ x^2(x^2 + 2)=x^4 + 2x^2 $. Subtract from dividend:
$ (x^4 + 5x^3 - 4x^2 + x - 10)-(x^4 + 2x^2)= 5x^3 - 6x^2 + x - 10 $

## Step 2: Next division
Divide $ 5x^3 $ by $ x^2 $ to get $ 5x $. Multiply divisor by $ 5x $: $ 5x(x^2 + 2)=5x^3 + 10x $. Subtract:
$ (5x^3 - 6x^2 + x - 10)-(5x^3 + 10x)= -6x^2 - 9x - 10 $

## Step 3: Final division
Divide $ -6x^2 $ by $ x^2 $ to get $ -6 $. Multiply divisor by $ -6 $: $ -6(x^2 + 2)= -6x^2 - 12 $. Subtract:
$ (-6x^2 - 9x - 10)-(-6x^2 - 12)= -9x + 2 $

# Answer: $ x^2 + 5x - 6 + \frac{-9x + 2}{x^2 + 2} $

### Problem 20: $ f(x) - g(x) $ where $ f(x)=3x^2 + 5x - 12 $, $ g(x)=x - 4 $
# Explanation:
## Step 1: Substitute the functions
$ f(x) - g(x)=(3x^2 + 5x - 12)-(x - 4) $

## Step 2: Distribute the negative sign
$ 3x^2 + 5x - 12 - x + 4 $

## Step 3: Combine like terms
$ 3x^2 + (5x - x) + (-12 + 4)= 3x^2 + 4x - 8 $

# Answer: $ 3x^2 + 4x - 8 $

### Problem 21: $ (7x^4 + 3x^3 + 4x^2 - 13) \div (x + 7) $
# Explanation:
## Step 1: Leading term division
Divide $ 7x^4 $ by $ x $ to get $ 7x^3 $. Multiply divisor by $ 7x^3 $: $ 7x^3(x + 7)=7x^4 + 49x^3 $. Subtract:
$ (7x^4 + 3x^3 + 4x^2 - 13)-(7x^4 + 49x^3)= -46x^3 + 4x^2 - 13 $

## Step 2: Next term
Divide $ -46x^3 $ by $ x $ to get $ -46x^2 $. Multiply divisor by $ -46x^2 $: $ -46x^2(x + 7)= -46x^3 - 322x^2 $. Subtract:
$ (-46x^3 + 4x^2 - 13)-(-46x^3 - 322x^2)= 326x^2 - 13 $

## Step 3: Next term
Divide $ 326x^2 $ by $ x $ to get $ 326x $. Multiply divisor by $ 326x $: $ 326x(x + 7)= 326x^2 + 2282x $. Subtract:
$ (326x^2 - 13)-(326x^2 + 2282x)= -2282x - 13 $

## Step 4: Last term
Divide $ -2282x $ by $ x $ to get $ -2282 $. Multiply divisor by $ -2282 $: $ -2282(x + 7)= -2282x - 15974 $. Subtract:
$ (-2282x - 13)-(-2282x - 15974)= 15961 $

# Answer: $ 7x^3 - 46x^2 + 326x - 2282 + \frac{15961}{x + 7} $

### Problem 22: $ (x^4 + 9x^3 - 2x^2 + x - 11) \div (x^2 + 3) $
# Explanation:
## Step 1: Divide leading terms
Divide $ x^4 $ by $ x^2 $ to get $ x^2 $. Multiply divisor by $ x^2 $: $ x^2(x^2 + 3)=x^4 + 3x^2 $. Subtract:
$ (x^4 + 9x^3 - 2x^2 + x - 11)-(x^4 + 3x^2)= 9x^3 - 5x^2 + x - 11 $

## Step 2: Next division
Divide $ 9x^3 $ by $ x^2 $ to get $ 9x $. Multiply divisor by $ 9x $: $ 9x(x^2 + 3)=9x^3 + 27x $. Subtract:
$ (9x^3 - 5x^2 + x - 11)-(9x^3 + 27x)= -5x^2 - 26x - 11 $

## Step 3: Final division
Divide $ -5x^2 $ by $ x^2 $ to get $ -5 $. Multiply divisor by $ -5 $: $ -5(x^2 + 3)= -5x^2 - 15 $. Subtract:
$ (-5x^2 - 26x - 11)-(-5x^2 - 15)= -26x + 4 $

# Answer: $ x^2 + 9x - 5 + \frac{-26x + 4}{x^2 + 3} $

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QUESTION IMAGE

Question

Explanation:

Problem 18 (assuming typo, let's take \( (6x^4 + 4x^3 + 7x^2 - 10) \div (x + 3) \))

Step 1: Set up polynomial long division

Step 2: Continue the division

Step 3: Next term

Step 4: Last term

Step 1: Divide leading terms

Step 2: Next division

Step 3: Final division

Step 1: Substitute the functions

Step 2: Distribute the negative sign

Step 3: Combine like terms

Answer:

Problem 19: \( (x^4 + 5x^3 - 4x^2 + x - 10) \div (x^2 + 2) \)