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

Problem: Dividing Polynomials

We are tasked with dividing each polynomial and expressing any remainders in fractional form. Let's solve each problem step by step.

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#### 1) \( z + 8 \big) z^2 - 19z - 11 \)

Step 1: Set up the division.
We divide \( z^2 - 19z - 11 \) by \( z + 8 \).

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{z^2}{z} = z
\]
So, the first term of the quotient is \( z \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
(z + 8)(z) = z^2 + 8z
\]
Subtract:
\[
(z^2 - 19z - 11) - (z^2 + 8z) = -27z - 11
\]

Step 4: Repeat the process with the new polynomial.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{-27z}{z} = -27
\]
So, the next term of the quotient is \( -27 \).

Multiply the entire divisor by this term and subtract:
\[
(z + 8)(-27) = -27z - 216
\]
Subtract:
\[
(-27z - 11) - (-27z - 216) = 205
\]

Step 5: Write the final answer.
The quotient is \( z - 27 \) and the remainder is \( 205 \). Express the remainder as a fraction:
\[
\frac{205}{z + 8}
\]

Final Answer:
\[
\boxed{z - 27 + \frac{205}{z + 8}}
\]

---

#### 2) \( h - 5 \big) -3h^2 + 15 \)

Step 1: Set up the division.
We divide \( -3h^2 + 15 \) by \( h - 5 \).

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{-3h^2}{h} = -3h
\]
So, the first term of the quotient is \( -3h \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
(h - 5)(-3h) = -3h^2 + 15h
\]
Subtract:
\[
(-3h^2 + 15) - (-3h^2 + 15h) = -15h + 15
\]

Step 4: Repeat the process with the new polynomial.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{-15h}{h} = -15
\]
So, the next term of the quotient is \( -15 \).

Multiply the entire divisor by this term and subtract:
\[
(h - 5)(-15) = -15h + 75
\]
Subtract:
\[
(-15h + 15) - (-15h + 75) = -60
\]

Step 5: Write the final answer.
The quotient is \( -3h - 15 \) and the remainder is \( -60 \). Express the remainder as a fraction:
\[
\frac{-60}{h - 5}
\]

Final Answer:
\[
\boxed{-3h - 15 - \frac{60}{h - 5}}
\]

---

#### 3) \( h + 3 \big) 2h^3 + 13h^2 - 14h + 4 \)

Step 1: Set up the division.
We divide \( 2h^3 + 13h^2 - 14h + 4 \) by \( h + 3 \).

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{2h^3}{h} = 2h^2
\]
So, the first term of the quotient is \( 2h^2 \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
(h + 3)(2h^2) = 2h^3 + 6h^2
\]
Subtract:
\[
(2h^3 + 13h^2 - 14h + 4) - (2h^3 + 6h^2) = 7h^2 - 14h + 4
\]

Step 4: Repeat the process with the new polynomial.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{7h^2}{h} = 7h
\]
So, the next term of the quotient is \( 7h \).

Multiply the entire divisor by this term and subtract:
\[
(h + 3)(7h) = 7h^2 + 21h
\]
Subtract:
\[
(7h^2 - 14h + 4) - (7h^2 + 21h) = -35h + 4
\]

Step 5: Repeat the process again.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{-35h}{h} = -35
\]
So, the next term of the quotient is \( -35 \).

Multiply the entire divisor by this term and subtract:
\[
(h + 3)(-35) = -35h - 105
\]
Subtract:
\[
(-35h + 4) - (-35h - 105) = 109
\]

Step 6: Write the final answer.
The quotient is \( 2h^2 + 7h - 35 \) and the remainder is \( 109 \). Express the remainder as a fraction:
\[
\frac{109}{h + 3}
\]

Final Answer:
\[
\boxed{2h^2 + 7h - 35 + \frac{109}{h + 3}}
\]

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#### 4) \( b + 4 \big) -b^3 + 16b^2 - 13b + 5 \)

Step 1: Set up the division.
We divide \( -b^3 + 16b^2 - 13b + 5 \) by \( b + 4 \).

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{-b^3}{b} = -b^2
\]
So, the first term of the quotient is \( -b^2 \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
(b + 4)(-b^2) = -b^3 - 4b^2
\]
Subtract:
\[
(-b^3 + 16b^2 - 13b + 5) - (-b^3 - 4b^2) = 20b^2 - 13b + 5
\]

Step 4: Repeat the process with the new polynomial.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{20b^2}{b} = 20b
\]
So, the next term of the quotient is \( 20b \).

Multiply the entire divisor by this term and subtract:
\[
(b + 4)(20b) = 20b^2 + 80b
\]
Subtract:
\[
(20b^2 - 13b + 5) - (20b^2 + 80b) = -93b + 5
\]

Step 5: Repeat the process again.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{-93b}{b} = -93
\]
So, the next term of the quotient is \( -93 \).

Multiply the entire divisor by this term and subtract:
\[
(b + 4)(-93) = -93b - 372
\]
Subtract:
\[
(-93b + 5) - (-93b - 372) = 377
\]

Step 6: Write the final answer.
The quotient is \( -b^2 + 20b - 93 \) and the remainder is \( 377 \). Express the remainder as a fraction:
\[
\frac{377}{b + 4}
\]

Final Answer:
\[
\boxed{-b^2 + 20b - 93 + \frac{377}{b + 4}}
\]

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#### 5) \( k - 9 \big) 3k^2 + 10k - 9 \)

Step 1: Set up the division.
We divide \( 3k^2 + 10k - 9 \) by \( k - 9 \).

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{3k^2}{k} = 3k
\]
So, the first term of the quotient is \( 3k \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
(k - 9)(3k) = 3k^2 - 27k
\]
Subtract:
\[
(3k^2 + 10k - 9) - (3k^2 - 27k) = 37k - 9
\]

Step 4: Repeat the process with the new polynomial.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{37k}{k} = 37
\]
So, the next term of the quotient is \( 37 \).

Multiply the entire divisor by this term and subtract:
\[
(k - 9)(37) = 37k - 333
\]
Subtract:
\[
(37k - 9) - (37k - 333) = 324
\]

Step 5: Write the final answer.
The quotient is \( 3k + 37 \) and the remainder is \( 324 \). Express the remainder as a fraction:
\[
\frac{324}{k - 9}
\]

Final Answer:
\[
\boxed{3k + 37 + \frac{324}{k - 9}}
\]

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#### 6) \( y - 4 \big) -3y^2 + 6 \)

Step 1: Set up the division.
We divide \( -3y^2 + 6 \) by \( y - 4 \).

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{-3y^2}{y} = -3y
\]
So, the first term of the quotient is \( -3y \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
(y - 4)(-3y) = -3y^2 + 12y
\]
Subtract:
\[
(-3y^2 + 6) - (-3y^2 + 12y) = -12y + 6
\]

Step 4: Repeat the process with the new polynomial.
Divide the leading term of the new polynomial by the leading term of the divisor:
\[
\frac{-12y}{y} = -12
\]
So, the next term of the quotient is \( -12 \).

Multiply the entire divisor by this term and subtract:
\[
(y - 4)(-12) = -12y + 48
\]
Subtract:
\[
(-12y + 6) - (-12y + 48) = -42
\]

Step 5: Write the final answer.
The quotient is \( -3y - 12 \) and the remainder is \( -42 \). Express the remainder as a fraction:
\[
\frac{-42}{y - 4}
\]

Final Answer:
\[
\boxed{-3y - 12 - \frac{42}{y - 4}}
\]

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Final Answers:

1. \(\boxed{z - 27 + \frac{205}{z + 8}}\)
2. \(\boxed{-3h - 15 - \frac{60}{h - 5}}\)
3. \(\boxed{2h^2 + 7h - 35 + \frac{109}{h + 3}}\)
4. \(\boxed{-b^2 + 20b - 93 + \frac{377}{b + 4}}\)
5. \(\boxed{3k + 37 + \frac{324}{k - 9}}\)
6. \(\boxed{-3y - 12 - \frac{42}{y - 4}}\)