Class 8 Division of Polynomials worksheet from Edugain.com, featuring problems and solutions for polynomial division.

Let's solve each problem step by step from the worksheet titled "Division of Polynomials" for Class 8.

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(1) Solve the following division of polynomials

We are given four polynomial divisions. We will simplify each one using factoring and basic algebraic identities.

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#### A) $ \frac{110a^3b^3 + 33b}{10a^3b^2 + 3} $

Factor numerator and denominator:

Numerator:
$ 110a^3b^3 + 33b = 11b(10a^3b^2 + 3) $

Denominator:
$ 10a^3b^2 + 3 $

So,

$$
\frac{11b(10a^3b^2 + 3)}{10a^3b^2 + 3} = 11b
$$

✔ Answer: $ 11b $

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#### B) $ \frac{48x + 108y^2}{12} $

Divide each term by 12:

$$
\frac{48x}{12} + \frac{108y^2}{12} = 4x + 9y^2
$$

✔ Answer: $ 4x + 9y^2 $

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#### C) $ \frac{22pq^2 + 66p^2q^3}{1 + 3pq} $

Factor numerator:

$ 22pq^2 + 66p^2q^3 = 22pq^2(1 + 3pq) $

Denominator: $ 1 + 3pq $

So,

$$
\frac{22pq^2(1 + 3pq)}{1 + 3pq} = 22pq^2
$$

✔ Answer: $ 22pq^2 $

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#### D) $ \frac{60p^2 + 36q}{5p^2 + 3q} $

Factor numerator:

$ 60p^2 + 36q = 12(5p^2 + 3q) $

Denominator: $ 5p^2 + 3q $

So,

$$
\frac{12(5p^2 + 3q)}{5p^2 + 3q} = 12
$$

✔ Answer: $ 12 $

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(2) $ (5y^2 - 4xy + 10y - 8x) \div (-4x + 5y) $

Let’s factor the numerator:
$ 5y^2 - 4xy + 10y - 8x $

Group terms:

$$
(5y^2 - 4xy) + (10y - 8x) = y(5y - 4x) + 2(5y - 4x)
$$

Now factor:

$$
(y + 2)(5y - 4x)
$$

Note: $ -4x + 5y = 5y - 4x $

So,

$$
\frac{(y + 2)(5y - 4x)}{5y - 4x} = y + 2
$$

✔ Answer: $ y + 2 $

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(3) $ (81b^4 - 16) \div (3b - 2) $

This is a difference of squares:

$ 81b^4 - 16 = (9b^2)^2 - (4)^2 = (9b^2 - 4)(9b^2 + 4) $

But $ 9b^2 - 4 = (3b)^2 - (2)^2 = (3b - 2)(3b + 2) $

So:

$$
81b^4 - 16 = (3b - 2)(3b + 2)(9b^2 + 4)
$$

Now divide by $ (3b - 2) $:

$$
\frac{(3b - 2)(3b + 2)(9b^2 + 4)}{3b - 2} = (3b + 2)(9b^2 + 4)
$$

Check options:

- a. $ (3b + 2)(9b^2 - 4) $
- b. $ (3b - 2)(9b^2 + 4) $
- c. $ (3b - 2)(9b^2 + 4) $ — same as b
- d. $ (3b - 2)(9b^2 - 4) $

Our answer is $ (3b + 2)(9b^2 + 4) $, which is not listed exactly.

Wait! Let's recheck.

We have:
$$
81b^4 - 16 = (9b^2)^2 - 4^2 = (9b^2 - 4)(9b^2 + 4)
= (3b - 2)(3b + 2)(9b^2 + 4)
$$

So dividing by $ (3b - 2) $ gives:
$$
(3b + 2)(9b^2 + 4)
$$

But this option is not in the list.

Wait — maybe I misread the options.

Let me check again:

a. $ (3b + 2)(9b^2 - 4) $ → no
b. $ (3b - 2)(9b^2 + 4) $ → no
c. $ (3b - 2)(9b^2 + 4) $ → same as b
d. $ (3b - 2)(9b^2 - 4) $ → no

None match? Wait — perhaps we made a mistake.

Wait! The divisor is $ (3b - 2) $. But our expression has $ (3b - 2)(3b + 2)(9b^2 + 4) $

So result is $ (3b + 2)(9b^2 + 4) $

But that's not among choices.

Wait — maybe it's a typo?

Alternatively, could the question be asking to factor completely or choose the correct form?

Wait — let's look at option a: $ (3b + 2)(9b^2 - 4) $

But $ 9b^2 - 4 = (3b - 2)(3b + 2) $, so that would be $ (3b + 2)^2(3b - 2) $, not matching.

Wait — perhaps the correct answer is not listed, but let's double-check.

Wait — maybe the expression was meant to be divided and simplified, but perhaps we can try long division?

Let’s try dividing $ 81b^4 - 16 $ by $ 3b - 2 $

Use polynomial long division.

Divide $ 81b^4 + 0b^3 + 0b^2 + 0b - 16 $ by $ 3b - 2 $

Step 1: $ 81b^4 \div 3b = 27b^3 $

Multiply: $ 27b^3(3b - 2) = 81b^4 - 54b^3 $

Subtract: $ (81b^4 - 16) - (81b^4 - 54b^3) = 54b^3 $

Bring down next term: $ 54b^3 + 0b^2 $

Step 2: $ 54b^3 \div 3b = 18b^2 $

Multiply: $ 18b^2(3b - 2) = 54b^3 - 36b^2 $

Subtract: $ 54b^3 + 0b^2 - (54b^3 - 36b^2) = 36b^2 $

Bring down: $ 36b^2 + 0b $

Step 3: $ 36b^2 \div 3b = 12b $

Multiply: $ 12b(3b - 2) = 36b^2 - 24b $

Subtract: $ 36b^2 + 0b - (36b^2 - 24b) = 24b $

Bring down: $ 24b - 16 $

Step 4: $ 24b \div 3b = 8 $

Multiply: $ 8(3b - 2) = 24b - 16 $

Subtract: $ 24b - 16 - (24b - 16) = 0 $

So quotient is: $ 27b^3 + 18b^2 + 12b + 8 $

But this is not matching any choice.

But earlier we had factored: $ (3b - 2)(3b + 2)(9b^2 + 4) $, so dividing by $ (3b - 2) $ gives $ (3b + 2)(9b^2 + 4) $

Let’s expand $ (3b + 2)(9b^2 + 4) $:

= $ 3b(9b^2 + 4) + 2(9b^2 + 4) = 27b^3 + 12b + 18b^2 + 8 = 27b^3 + 18b^2 + 12b + 8 $

Which matches the long division result.

So the answer is $ 27b^3 + 18b^2 + 12b + 8 $

But none of the options match this.

Wait — the choices are:

a. $ (3b + 2)(9b^2 - 4) $ → $ (3b+2)(9b^2 - 4) = 27b^3 - 12b + 18b^2 - 8 = 27b^3 + 18b^2 - 12b - 8 $ → no

b. $ (3b - 2)(9b^2 + 4) $ → $ 27b^3 + 12b - 18b^2 - 8 = 27b^3 - 18b^2 + 12b - 8 $ → no

c. $ (3b - 2)(9b^2 + 4) $ → same as b

d. $ (3b - 2)(9b^2 - 4) $ → $ 27b^3 - 12b - 18b^2 + 8 = 27b^3 - 18b^2 - 12b + 8 $

None match.

Wait — but our correct answer is $ (3b + 2)(9b^2 + 4) $

But this is not listed.

However, notice that $ (3b + 2)(9b^2 + 4) $ is not among options.

But wait — maybe the original expression is $ 81b^4 - 16 $, and divisor is $ 3b - 2 $, so the answer should be $ (3b + 2)(9b^2 + 4) $

But since this isn't an option, perhaps there is a typo.

Wait — look at option a: $ (3b + 2)(9b^2 - 4) $

But $ 9b^2 - 4 = (3b - 2)(3b + 2) $, so $ (3b + 2)(9b^2 - 4) = (3b + 2)^2(3b - 2) $

That’s not right.

Wait — perhaps the correct answer is not listed, but let's see if any of them equal $ (3b + 2)(9b^2 + 4) $

No.

But maybe the question is multiple choice, and only one is correct, but all seem wrong.

Wait — perhaps I made a mistake.

Wait — $ 81b^4 - 16 = (9b^2)^2 - 4^2 = (9b^2 - 4)(9b^2 + 4) $

And $ 9b^2 - 4 = (3b - 2)(3b + 2) $

So $ 81b^4 - 16 = (3b - 2)(3b + 2)(9b^2 + 4) $

So $ \frac{81b^4 - 16}{3b - 2} = (3b + 2)(9b^2 + 4) $

Now, is this equal to any option?

No.

But look — option a is $ (3b + 2)(9b^2 - 4) $ — close but not same.

Wait — unless the question is $ (81b^4 - 16) \div (3b + 2) $? Then it would be $ (3b - 2)(9b^2 + 4) $, which is not listed either.

Hmm.

Wait — perhaps the correct answer is not among the choices, but let's move on and come back.

Maybe the question has a typo.

But let’s assume the intended answer is $ (3b + 2)(9b^2 + 4) $, but since it's not listed, perhaps we need to pick the closest.

But none are close.

Wait — let's check if option d is $ (3b - 2)(9b^2 - 4) $ — that's $ (3b - 2)^2(3b + 2) $ — no.

Alternatively, maybe the question is to factor, and the answer is $ (3b + 2)(9b^2 + 4) $, but it's not listed.

Perhaps the answer is not among the options, but let's skip and return later.

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(4) $ (b^2 - 3ab - 8b + 24a) \div (b - 8) $

Let’s factor numerator:

Group: $ (b^2 - 3ab) + (-8b + 24a) = b(b - 3a) -8(b - 3a) = (b - 8)(b - 3a) $

Wait: $ b(b - 3a) = b^2 - 3ab $, good

$ -8(b - 3a) = -8b + 24a $, yes!

So: $ (b - 8)(b - 3a) $

Divide by $ (b - 8) $: $ b - 3a $

Now check options:

a. $ 3a + b $
b. $ a - 3b $
c. $ -3a + b $ → same as $ b - 3a $ ✔
d. $ -3a - b $

So c. $ -3a + b $ is correct.

✔ Answer: c

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(5) $ (9a^2 + 18ab + 9b^2 - c^2) \div (3a + 3b + c) $

First, group terms:

$ 9a^2 + 18ab + 9b^2 = 9(a^2 + 2ab + b^2) = 9(a + b)^2 $

So expression becomes:

$ 9(a + b)^2 - c^2 $ — difference of squares!

= $ [3(a + b)]^2 - c^2 = (3a + 3b - c)(3a + 3b + c) $

So numerator = $ (3a + 3b - c)(3a + 3b + c) $

Divide by $ (3a + 3b + c) $: result is $ 3a + 3b - c $

Now check options:

a. $ (3a - 3b + c) $
b. $ (3a - 3b - c) $
c. $ (3a + 3b + c) $
d. $ (3a + 3b - c) $ ✔

✔ Answer: d

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(6) $ (64z^2 - 80z + 25) \div (8z - 5) $

First, factor numerator:

$ 64z^2 - 80z + 25 $

Check discriminant: $ 80^2 - 4*64*25 = 6400 - 6400 = 0 $ → perfect square

So $ = (8z - 5)^2 $

So $ \frac{(8z - 5)^2}{8z - 5} = 8z - 5 $

Now check options:

a. $ (5z - 8) $
b. $ (8z + 5) $
c. $ (8z - 5) $ ✔
d. $ (5z + 8) $

✔ Answer: c

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(7) $ (2x^2 + 21x + 27) \div (x + 9) $

Try factoring numerator:

Find two numbers multiplying to $ 2*27 = 54 $, adding to 21: 18 and 3

So:

$ 2x^2 + 18x + 3x + 27 = 2x(x + 9) + 3(x + 9) = (2x + 3)(x + 9) $

So $ \frac{(2x + 3)(x + 9)}{x + 9} = 2x + 3 $

Check options:

a. $ 2x - 3 $
b. $ -2x + 3 $
c. $ 3x + 2 $
d. $ 2x + 3 $ ✔

✔ Answer: d

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(8) $ (xy + 5y - 7x - 35) \div (x + 5) $

Group terms:

$ (xy + 5y) + (-7x - 35) = y(x + 5) -7(x + 5) = (y - 7)(x + 5) $

So $ \frac{(y - 7)(x + 5)}{x + 5} = y - 7 $

Check options:

a. $ y - 7 $ ✔
b. $ y + 7 $
c. $ -y - 7 $
d. $ -y + 1 $

✔ Answer: a

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Now go back to (3)

We had:

$ (81b^4 - 16) \div (3b - 2) = (3b + 2)(9b^2 + 4) $

But this is not among the options.

Wait — perhaps the question is $ (81b^4 - 16) \div (3b + 2) $? Then answer would be $ (3b - 2)(9b^2 + 4) $

But that’s not listed either.

Wait — look at option a: $ (3b + 2)(9b^2 - 4) $

But $ 9b^2 - 4 = (3b - 2)(3b + 2) $, so this is $ (3b + 2)^2(3b - 2) $, which is not correct.

Wait — unless the expression is different.

Wait — perhaps the original expression is $ 81b^4 - 16 $, and divisor is $ 3b - 2 $, and the answer is $ 27b^3 + 18b^2 + 12b + 8 $, but none of the choices match.

But look at option a: $ (3b + 2)(9b^2 - 4) $

Expand: $ 3b(9b^2 - 4) + 2(9b^2 - 4) = 27b^3 - 12b + 18b^2 - 8 = 27b^3 + 18b^2 - 12b - 8 $

Not matching.

But our correct expansion is $ 27b^3 + 18b^2 + 12b + 8 $

So no option matches.

Wait — maybe the divisor is $ 3b + 2 $? Then:

$ \frac{81b^4 - 16}{3b + 2} = (3b - 2)(9b^2 + 4) $

Now check if any option is $ (3b - 2)(9b^2 + 4) $

Yes! Options b and c are both $ (3b - 2)(9b^2 + 4) $

Wait — in the image:

b. $ (3b - 2)(9b^2 + 4) $

c. $ (3b - 2)(9b^2 + 4) $

Same thing? Probably typo.

But in the image, b and c are identical — likely a formatting error.

But if the divisor were $ 3b + 2 $, then answer is $ (3b - 2)(9b^2 + 4) $, which is b or c.

But the question says $ (3b - 2) $, so divisor is $ 3b - 2 $, so answer should be $ (3b + 2)(9b^2 + 4) $, which is not listed.

So either:

- There is a typo in the question or options
- Or we must accept that none are correct

But since the problem asks to "choose correct answer(s)", and only b and c are the same and plausible, perhaps the divisor was meant to be $ 3b + 2 $, or the expression is different.

Wait — another possibility: maybe the expression is $ (81b^4 - 16) \div (3b - 2) $, and the answer is $ (3b + 2)(9b^2 + 4) $, but since it's not listed, perhaps the intended answer is a, but it's not.

Alternatively, perhaps the expression is $ (81b^4 - 16) \div (3b - 2) $, and they want us to recognize it as $ (9b^2 - 4)(9b^2 + 4) / (3b - 2) $, but still.

Given the discrepancy, and since b and c are identical, and both are $ (3b - 2)(9b^2 + 4) $, which is close but not correct.

Wait — unless the question is to factor, and the answer is $ (3b - 2)(9b^2 + 4) $, but that's the numerator, not the quotient.

I think there might be a typo in the problem or options.

But based on correct math:

✔ For (3), correct answer is $ (3b + 2)(9b^2 + 4) $, which is not listed.

But if we assume the divisor is $ 3b + 2 $, then answer is $ (3b - 2)(9b^2 + 4) $, which is b or c.

Since b and c are the same, and a and d are different, perhaps the intended answer is b or c.

But based on the given divisor $ (3b - 2) $, the answer is not listed.

However, looking back at the image, perhaps I misread.

Wait — in the image, b and c are both written as $ (3b - 2)(9b^2 + 4) $? That seems like a duplication.

Possibly, the correct answer is b or c, assuming typo in the divisor.

But strictly speaking, with divisor $ (3b - 2) $, answer is $ (3b + 2)(9b^2 + 4) $, not listed.

So perhaps the question has a typo.

But for now, let’s summarize what we have.

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Final Answers:

(1)
A) $ 11b $
B) $ 4x + 9y^2 $
C) $ 22pq^2 $
D) $ 12 $

(2) $ y + 2 $

(3) Not listed — should be $ (3b + 2)(9b^2 + 4) $, but if forced to choose, possibly b or c if divisor was $ 3b + 2 $

But since b and c are identical, and likely a typo, and a is $ (3b + 2)(9b^2 - 4) $, which is wrong.

Wait — perhaps the answer is a? No.

Alternatively, maybe the expression is $ (81b^4 - 16) \div (3b - 2) $, and the answer is $ (3b + 2)(9b^2 + 4) $, and they expect us to write it, but since it's multiple choice, and none match, perhaps it's a flaw.

But let’s assume the intended answer is a — no.

Another idea: maybe the expression is $ (81b^4 - 16) \div (3b - 2) $, and they want the quotient as $ (3b + 2)(9b^2 + 4) $, but since it's not listed, and b and c are the same, perhaps the answer is b.

But I think the most likely explanation is that the divisor is $ 3b + 2 $, and the answer is $ (3b - 2)(9b^2 + 4) $, which is b or c.

Given that, and since b and c are identical, we can say b or c.

But in the image, both b and c are $ (3b - 2)(9b^2 + 4) $ — probably a copy-paste error.

So likely, answer is b or c.

But to be safe, let’s assume the intended answer is b.

But strictly, with divisor $ 3b - 2 $, answer is not listed.

But since the rest are correct, and this one is problematic, we'll note it.

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Summary of Answers:

| Question | Answer |
|--------|--------|
| (1) A | $ 11b $ |
| (1) B | $ 4x + 9y^2 $ |
| (1) C | $ 22pq^2 $ |
| (1) D | $ 12 $ |
| (2) | $ y + 2 $ |
| (3) | Not listed — should be $ (3b + 2)(9b^2 + 4) $; if forced, possibly b or c (but both same) |
| (4) | c. $ -3a + b $ |
| (5) | d. $ (3a + 3b - c) $ |
| (6) | c. $ (8z - 5) $ |
| (7) | d. $ 2x + 3 $ |
| (8) | a. $ y - 7 $ |

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Final Boxed Answers:

(1)
- A) $ \boxed{11b} $
- B) $ \boxed{4x + 9y^2} $
- C) $ \boxed{22pq^2} $
- D) $ \boxed{12} $

(2) $ \boxed{y + 2} $

(3) $ \boxed{\text{Not listed}} $ — correct is $ (3b + 2)(9b^2 + 4) $

(4) $ \boxed{\text{c}} $

(5) $ \boxed{\text{d}} $

(6) $ \boxed{\text{c}} $

(7) $ \boxed{\text{d}} $

(8) $ \boxed{\text{a}} $

> Note: For question (3), the correct answer is not among the choices due to a possible typo in the options or divisor. If the divisor were $ 3b + 2 $, then the answer would be $ (3b - 2)(9b^2 + 4) $, which is option b or c.