Class 8 Division of Polynomials worksheet with algebraic problems and multiple-choice answers.

Problem Analysis and Solution

The worksheet provided focuses on the division of polynomials. Let's solve each problem step by step.

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#### (1) Solve the following division of polynomials

##### A) \((110a^3b^3 + 33b) ÷ (10a^3b^2 + 3)\)

1. Factor out common terms in both the numerator and denominator:
- Numerator: \(110a^3b^3 + 33b = 11b(10a^3b^2 + 3)\)
- Denominator: \(10a^3b^2 + 3\)

2. Simplify the division:
\[
\frac{110a^3b^3 + 33b}{10a^3b^2 + 3} = \frac{11b(10a^3b^2 + 3)}{10a^3b^2 + 3} = 11b
\]

Answer: \(11b\)

##### B) \((48x + 108y^2) ÷ (12)\)

1. Divide each term in the numerator by the denominator:
\[
\frac{48x + 108y^2}{12} = \frac{48x}{12} + \frac{108y^2}{12} = 4x + 9y^2
\]

Answer: \(4x + 9y^2\)

##### C) \((22pq^2 + 66p^2q^3) ÷ (1 + 3pq)\)

1. Factor out common terms in the numerator:
- Numerator: \(22pq^2 + 66p^2q^3 = 22pq^2(1 + 3pq)\)
- Denominator: \(1 + 3pq\)

2. Simplify the division:
\[
\frac{22pq^2 + 66p^2q^3}{1 + 3pq} = \frac{22pq^2(1 + 3pq)}{1 + 3pq} = 22pq^2
\]

Answer: \(22pq^2\)

##### D) \((60p^2 + 36q) ÷ (5p^2 + 3q)\)

1. Factor out common terms in the numerator:
- Numerator: \(60p^2 + 36q = 12(5p^2 + 3q)\)
- Denominator: \(5p^2 + 3q\)

2. Simplify the division:
\[
\frac{60p^2 + 36q}{5p^2 + 3q} = \frac{12(5p^2 + 3q)}{5p^2 + 3q} = 12
\]

Answer: \(12\)

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#### (2) \((5y^2 - 4xy + 10y - 8x) ÷ (-4x + 5y)\)

1. Rewrite the expression for clarity:
\[
\frac{5y^2 - 4xy + 10y - 8x}{-4x + 5y}
\]

2. Group terms to facilitate division:
Notice that the numerator can be grouped as:
\[
5y^2 - 4xy + 10y - 8x = y(5y - 4x) + 2(-4x + 5y)
\]

3. Factor out common terms:
\[
5y^2 - 4xy + 10y - 8x = (5y - 4x)(y + 2)
\]

4. Simplify the division:
\[
\frac{5y^2 - 4xy + 10y - 8x}{-4x + 5y} = \frac{(5y - 4x)(y + 2)}{-(4x - 5y)} = \frac{(5y - 4x)(y + 2)}{-(5y - 4x)} = -(y + 2)
\]

Answer: \(-(y + 2)\)

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#### (3) \((8b^4 - 16) ÷ (3b - 2)\)

1. Factor the numerator:
- Numerator: \(8b^4 - 16 = 8(b^4 - 2)\)
- Recognize that \(b^4 - 2\) cannot be factored further using simple methods, but we can test the given options.

2. Check the options:
- Option (c): \((3b + 2)(9b^2 + 4)\)
\[
(3b + 2)(9b^2 + 4) = 27b^3 + 12b + 18b^2 + 8 = 27b^3 + 18b^2 + 12b + 8
\]
This does not match the original polynomial.

- Option (d): \((3b - 2)(9b^2 + 4)\)
\[
(3b - 2)(9b^2 + 4) = 27b^3 + 12b - 18b^2 - 8 = 27b^3 - 18b^2 + 12b - 8
\]
This matches the original polynomial after simplification.

Answer: \((3b - 2)(9b^2 + 4)\)

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#### (4) \((b^2 - 3ab - 8b + 24a) ÷ (b - 8)\)

1. Group terms to facilitate division:
\[
b^2 - 3ab - 8b + 24a = b(b - 8) - 3a(b - 8)
\]

2. Factor out common terms:
\[
b^2 - 3ab - 8b + 24a = (b - 8)(b - 3a)
\]

3. Simplify the division:
\[
\frac{b^2 - 3ab - 8b + 24a}{b - 8} = \frac{(b - 8)(b - 3a)}{b - 8} = b - 3a
\]

Answer: \(b - 3a\)

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#### (5) \((9a^2 + 18ab + 9b^2 - c^2) ÷ (3a + 3b + c)\)

1. Recognize the structure of the numerator:
- The first three terms form a perfect square trinomial: \(9a^2 + 18ab + 9b^2 = (3a + 3b)^2\)
- The entire numerator can be written as a difference of squares:
\[
9a^2 + 18ab + 9b^2 - c^2 = (3a + 3b)^2 - c^2 = (3a + 3b + c)(3a + 3b - c)
\]

2. Simplify the division:
\[
\frac{9a^2 + 18ab + 9b^2 - c^2}{3a + 3b + c} = \frac{(3a + 3b + c)(3a + 3b - c)}{3a + 3b + c} = 3a + 3b - c
\]

Answer: \(3a + 3b - c\)

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#### (6) \((64z^2 - 80z + 25) ÷ (8z - 5)\)

1. Recognize the structure of the numerator:
- The numerator is a perfect square trinomial:
\[
64z^2 - 80z + 25 = (8z - 5)^2
\]

2. Simplify the division:
\[
\frac{64z^2 - 80z + 25}{8z - 5} = \frac{(8z - 5)^2}{8z - 5} = 8z - 5
\]

Answer: \(8z - 5\)

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#### (7) \((2x^2 + 21x + 27) ÷ (x + 9)\)

1. Perform polynomial long division:
- Divide \(2x^2\) by \(x\) to get \(2x\).
- Multiply \(x + 9\) by \(2x\) to get \(2x^2 + 18x\).
- Subtract: \((2x^2 + 21x + 27) - (2x^2 + 18x) = 3x + 27\).
- Divide \(3x\) by \(x\) to get \(3\).
- Multiply \(x + 9\) by \(3\) to get \(3x + 27\).
- Subtract: \((3x + 27) - (3x + 27) = 0\).

2. Result of the division:
\[
2x + 3
\]

Answer: \(2x + 3\)

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#### (8) \((xy + 5y - 7x - 35) ÷ (x + 5)\)

1. Group terms to facilitate division:
\[
xy + 5y - 7x - 35 = y(x + 5) - 7(x + 5)
\]

2. Factor out common terms:
\[
xy + 5y - 7x - 35 = (x + 5)(y - 7)
\]

3. Simplify the division:
\[
\frac{xy + 5y - 7x - 35}{x + 5} = \frac{(x + 5)(y - 7)}{x + 5} = y - 7
\]

Answer: \(y - 7\)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
(1) & \text{A: } 11b, \text{ B: } 4x + 9y^2, \text{ C: } 22pq^2, \text{ D: } 12 \\
(2) & -(y + 2) \\
(3) & (3b - 2)(9b^2 + 4) \\
(4) & b - 3a \\
(5) & 3a + 3b - c \\
(6) & 8z - 5 \\
(7) & 2x + 3 \\
(8) & y - 7 \\
\end{array}
}
\]