Printable worksheet featuring six polynomial long division problems designed to help students practice dividing expressions and handling remainders.

It looks like you've uploaded a Polynomial Long Division Worksheet, and the task is to divide each polynomial using long division and write the remainder in fractional form.

Let’s go through each problem one by one. I'll solve them step-by-step, showing the long division process and then express the remainder as a fraction.

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🔷 Problem 1:

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

We perform polynomial long division:

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 $-9$: now we have $37k - 9$
5. Divide $37k \div k = 37$
6. Multiply: $37(k - 9) = 37k - 333$
7. Subtract: $(37k - 9) - (37k - 333) = 324$

So:
$$
\frac{3k^2 + 10k - 9}{k - 9} = 3k + 37 + \frac{324}{k - 9}
$$

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

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🔷 Problem 2:

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

Note: The dividend is $-3y^2 + 0y + 6$, so we include the missing $y$ term.

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 $+6$: now $-12y + 6$
5. Divide $-12y \div y = -12$
6. Multiply: $-12(y - 4) = -12y + 48$
7. Subtract: $(-12y + 6) - (-12y + 48) = -42$

So:
$$
\frac{-3y^2 + 6}{y - 4} = -3y - 12 + \frac{-42}{y - 4}
$$

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

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🔷 Problem 3:

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

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:
$$
\frac{z^2 - 19z - 11}{z + 8} = z - 27 + \frac{205}{z + 8}
$$

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

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🔷 Problem 4:

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

Wait — there's an error here: $-b^2 + 16b^2 = 15b^2$, so the dividend is:

$$
15b^2 - 13b + 5
$$

Now divide:
$$
\frac{15b^2 - 13b + 5}{b + 4}
$$

1. $15b^2 \div b = 15b$
2. Multiply: $15b(b + 4) = 15b^2 + 60b$
3. Subtract: $(15b^2 - 13b) - (15b^2 + 60b) = -73b$
4. Bring down $+5$: $-73b + 5$
5. $-73b \div b = -73$
6. Multiply: $-73(b + 4) = -73b - 292$
7. Subtract: $(-73b + 5) - (-73b - 292) = 297$

So:
$$
\frac{15b^2 - 13b + 5}{b + 4} = 15b - 73 + \frac{297}{b + 4}
$$

✔ Answer: $15b - 73 + \frac{297}{b + 4}$

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🔷 Problem 5:

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

Wait — again, combine like terms: $2h^2 + 13h^2 = 15h^2$

So dividend is: $15h^2 - 14h + 4$

Now divide:
$$
\frac{15h^2 - 14h + 4}{h + 3}
$$

1. $15h^2 \div h = 15h$
2. Multiply: $15h(h + 3) = 15h^2 + 45h$
3. Subtract: $(15h^2 - 14h) - (15h^2 + 45h) = -59h$
4. Bring down $+4$: $-59h + 4$
5. $-59h \div h = -59$
6. Multiply: $-59(h + 3) = -59h - 177$
7. Subtract: $(-59h + 4) - (-59h - 177) = 181$

So:
$$
\frac{15h^2 - 14h + 4}{h + 3} = 15h - 59 + \frac{181}{h + 3}
$$

✔ Answer: $15h - 59 + \frac{181}{h + 3}$

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🔷 Problem 6:

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

This is identical to Problem 3!

So same result:

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

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

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $3k + 37 + \frac{324}{k - 9}$ |
| 2 | $-3y - 12 - \frac{42}{y - 4}$ |
| 3 | $z - 27 + \frac{205}{z + 8}$ |
| 4 | $15b - 73 + \frac{297}{b + 4}$ |
| 5 | $15h - 59 + \frac{181}{h + 3}$ |
| 6 | $z - 27 + \frac{205}{z + 8}$ |

> Note: Problems 3 and 6 are identical.

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