Dividing Polynomials worksheet with six problems requiring polynomial division and fractional remainders.

Here are the step-by-step solutions for each polynomial division problem. I will use synthetic division or long division logic to find the quotient and the remainder. The final answer will be written as:
Quotient + (Remainder / Divisor)

1) $(z^2 - 19z - 11) \div (z + 8)$

Step 1: Set up the division. We are dividing by $z + 8$, so we use $-8$ for synthetic division.
Coefficients of dividend: $1, -19, -11$

Step 2: Perform synthetic division.
* Bring down the $1$.
* Multiply $1 \times -8 = -8$.
* Add to next coefficient: $-19 + (-8) = -27$.
* Multiply $-27 \times -8 = 216$.
* Add to next coefficient: $-11 + 216 = 205$.

Step 3: Interpret results.
* Quotient coefficients: $1, -27$ $\rightarrow$ $z - 27$
* Remainder: $205$

Result: $z - 27 + \frac{205}{z + 8}$

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2) $(-3h^2 + 15) \div (h - 5)$

Note: The dividend is missing the $h$ term. We must include a placeholder $0h$. So, coefficients are $-3, 0, 15$.

Step 1: Set up synthetic division with $5$ (from $h - 5$).
Coefficients: $-3, 0, 15$

Step 2: Perform synthetic division.
* Bring down $-3$.
* Multiply $-3 \times 5 = -15$.
* Add to next coefficient: $0 + (-15) = -15$.
* Multiply $-15 \times 5 = -75$.
* Add to next coefficient: $15 + (-75) = -60$.

Step 3: Interpret results.
* Quotient coefficients: $-3, -15$ $\rightarrow$ $-3h - 15$
* Remainder: $-60$

Result: $-3h - 15 - \frac{60}{h - 5}$

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3) $(2h^3 + 13h^2 - 14h + 4) \div (h + 3)$

Step 1: Set up synthetic division with $-3$ (from $h + 3$).
Coefficients: $2, 13, -14, 4$

Step 2: Perform synthetic division.
* Bring down $2$.
* Multiply $2 \times -3 = -6$.
* Add: $13 + (-6) = 7$.
* Multiply $7 \times -3 = -21$.
* Add: $-14 + (-21) = -35$.
* Multiply $-35 \times -3 = 105$.
* Add: $4 + 105 = 109$.

Step 3: Interpret results.
* Quotient coefficients: $2, 7, -35$ $\rightarrow$ $2h^2 + 7h - 35$
* Remainder: $109$

Result: $2h^2 + 7h - 35 + \frac{109}{h + 3}$

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4) $(-b^3 + 16b^2 - 13b + 5) \div (b + 4)$

Step 1: Set up synthetic division with $-4$ (from $b + 4$).
Coefficients: $-1, 16, -13, 5$

Step 2: Perform synthetic division.
* Bring down $-1$.
* Multiply $-1 \times -4 = 4$.
* Add: $16 + 4 = 20$.
* Multiply $20 \times -4 = -80$.
* Add: $-13 + (-80) = -93$.
* Multiply $-93 \times -4 = 372$.
* Add: $5 + 372 = 377$.

Step 3: Interpret results.
* Quotient coefficients: $-1, 20, -93$ $\rightarrow$ $-b^2 + 20b - 93$
* Remainder: $377$

Result: $-b^2 + 20b - 93 + \frac{377}{b + 4}$

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5) $(3k^2 + 10k - 9) \div (k - 9)$

Step 1: Set up synthetic division with $9$ (from $k - 9$).
Coefficients: $3, 10, -9$

Step 2: Perform synthetic division.
* Bring down $3$.
* Multiply $3 \times 9 = 27$.
* Add: $10 + 27 = 37$.
* Multiply $37 \times 9 = 333$.
* Add: $-9 + 333 = 324$.

Step 3: Interpret results.
* Quotient coefficients: $3, 37$ $\rightarrow$ $3k + 37$
* Remainder: $324$

Result: $3k + 37 + \frac{324}{k - 9}$

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6) $(-3y^2 + 6) \div (y - 4)$

Note: The dividend is missing the $y$ term. Use placeholder $0y$. Coefficients: $-3, 0, 6$.

Step 1: Set up synthetic division with $4$ (from $y - 4$).
Coefficients: $-3, 0, 6$

Step 2: Perform synthetic division.
* Bring down $-3$.
* Multiply $-3 \times 4 = -12$.
* Add: $0 + (-12) = -12$.
* Multiply $-12 \times 4 = -48$.
* Add: $6 + (-48) = -42$.

Step 3: Interpret results.
* Quotient coefficients: $-3, -12$ $\rightarrow$ $-3y - 12$
* Remainder: $-42$

Result: $-3y - 12 - \frac{42}{y - 4}$

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Final Answer:

1) $z - 27 + \frac{205}{z + 8}$

2) $-3h - 15 - \frac{60}{h - 5}$

3) $2h^2 + 7h - 35 + \frac{109}{h + 3}$

4) $-b^2 + 20b - 93 + \frac{377}{b + 4}$

5) $3k + 37 + \frac{324}{k - 9}$

6) $-3y - 12 - \frac{42}{y - 4}$