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

Let's solve each polynomial division problem step by step using long division. We'll divide each polynomial and express any remainder as a fraction.

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1) Divide:

$$
\frac{z^2 - 19z - 11}{z + 8}
$$

We perform polynomial long division:

1. Divide $ z^2 \div z = z $
2. Multiply: $ z(z + 8) = z^2 + 8z $
3. Subtract: $ (z^2 - 19z) - (z^2 + 8z) = -27z $
4. Bring down $-11$: $ -27z - 11 $
5. Divide $ -27z \div z = -27 $
6. Multiply: $ -27(z + 8) = -27z - 216 $
7. Subtract: $ (-27z - 11) - (-27z - 216) = 205 $

So, the result is:
$$
z - 27 + \frac{205}{z + 8}
$$

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

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2) Divide:

$$
\frac{-3h^2 + 15}{h - 5}
$$

Note: The dividend is $ -3h^2 + 0h + 15 $

1. Divide $ -3h^2 \div h = -3h $
2. Multiply: $ -3h(h - 5) = -3h^2 + 15h $
3. Subtract: $ (-3h^2 + 0h) - (-3h^2 + 15h) = -15h $
4. Bring down $+15$: $ -15h + 15 $
5. Divide $ -15h \div h = -15 $
6. Multiply: $ -15(h - 5) = -15h + 75 $
7. Subtract: $ (-15h + 15) - (-15h + 75) = -60 $

So, the result is:
$$
-3h - 15 + \frac{-60}{h - 5} = -3h - 15 - \frac{60}{h - 5}
$$

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

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3) Divide:

$$
\frac{2h^3 + 13h^2 - 14h + 4}{h + 3}
$$

1. Divide $ 2h^3 \div h = 2h^2 $
2. Multiply: $ 2h^2(h + 3) = 2h^3 + 6h^2 $
3. Subtract: $ (2h^3 + 13h^2) - (2h^3 + 6h^2) = 7h^2 $
4. Bring down $-14h$: $ 7h^2 - 14h $
5. Divide $ 7h^2 \div h = 7h $
6. Multiply: $ 7h(h + 3) = 7h^2 + 21h $
7. Subtract: $ (7h^2 - 14h) - (7h^2 + 21h) = -35h $
8. Bring down $+4$: $ -35h + 4 $
9. Divide $ -35h \div h = -35 $
10. Multiply: $ -35(h + 3) = -35h - 105 $
11. Subtract: $ (-35h + 4) - (-35h - 105) = 109 $

So, the result is:
$$
2h^2 + 7h - 35 + \frac{109}{h + 3}
$$

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

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4) Divide:

$$
\frac{-b^3 + 16b^2 - 13b + 5}{b + 4}
$$

1. Divide $ -b^3 \div b = -b^2 $
2. Multiply: $ -b^2(b + 4) = -b^3 - 4b^2 $
3. Subtract: $ (-b^3 + 16b^2) - (-b^3 - 4b^2) = 20b^2 $
4. Bring down $-13b$: $ 20b^2 - 13b $
5. Divide $ 20b^2 \div b = 20b $
6. Multiply: $ 20b(b + 4) = 20b^2 + 80b $
7. Subtract: $ (20b^2 - 13b) - (20b^2 + 80b) = -93b $
8. Bring down $+5$: $ -93b + 5 $
9. Divide $ -93b \div b = -93 $
10. Multiply: $ -93(b + 4) = -93b - 372 $
11. Subtract: $ (-93b + 5) - (-93b - 372) = 377 $

So, the result is:
$$
-b^2 + 20b - 93 + \frac{377}{b + 4}
$$

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

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5) Divide:

$$
\frac{3k^2 + 10k - 9}{k - 9}
$$

1. Divide $ 3k^2 \div k = 3k $
2. Multiply: $ 3k(k - 9) = 3k^2 - 27k $
3. Subtract: $ (3k^2 + 10k) - (3k^2 - 27k) = 37k $
4. Bring down $-9$: $ 37k - 9 $
5. Divide $ 37k \div k = 37 $
6. Multiply: $ 37(k - 9) = 37k - 333 $
7. Subtract: $ (37k - 9) - (37k - 333) = 324 $

So, the result is:
$$
3k + 37 + \frac{324}{k - 9}
$$

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

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6) Divide:

$$
\frac{-3y^2 + 6}{y - 4}
$$

Note: Dividend is $ -3y^2 + 0y + 6 $

1. Divide $ -3y^2 \div y = -3y $
2. Multiply: $ -3y(y - 4) = -3y^2 + 12y $
3. Subtract: $ (-3y^2 + 0y) - (-3y^2 + 12y) = -12y $
4. Bring down $+6$: $ -12y + 6 $
5. Divide $ -12y \div y = -12 $
6. Multiply: $ -12(y - 4) = -12y + 48 $
7. Subtract: $ (-12y + 6) - (-12y + 48) = -42 $

So, the result is:
$$
-3y - 12 + \frac{-42}{y - 4} = -3y - 12 - \frac{42}{y - 4}
$$

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

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✔ Final Answers:

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} $

Let me know if you'd like these checked with synthetic division or explained further!