Worksheet on Operations on Rational Algebraic Expressions | PDF - Free Printable

Let's solve each problem step by step from the worksheet titled "Algebraic Expressions".

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1. State the degrees of the following polynomials.

The degree of a polynomial is the highest power of the variable in the expression.

a. $ 5x $
→ Degree = 1 (since $ x^1 $)

b. $ 2x + 3y $
→ Both terms are degree 1, so the degree of the polynomial is 1

c. $ 5x^2 - 4x^3 + 2x + 1 $
→ The highest power of $ x $ is $ x^3 $, so degree = 3

✔ Answers:
a. 1
b. 1
c. 3

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2. Arrange the polynomials according to the descending order of their degrees.

We need to find the degree of each polynomial:

a. $ 4x^2 + 3x^4 - 7x^3 + 3x + 2 $
→ Highest power: $ x^4 $ → degree = 4

b. $ 5x^3 - x^2 - 7x + 4x + 8 $
→ Simplify: $ 5x^3 - x^2 - 3x + 8 $
→ Highest power: $ x^3 $ → degree = 3

So, in descending order of degree:
a (degree 4), b (degree 3)

✔ Answer:
a, then b

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3. The sides of a triangle are given by $ 3x^2 - y^2 $, $ 4x^2 - 7xy + 4y $, and $ -3x + 7xy + 8y^2 $. Find its perimeter.

Perimeter = sum of all sides.

Add the three expressions:

$$
(3x^2 - y^2) + (4x^2 - 7xy + 4y) + (-3x + 7xy + 8y^2)
$$

Now combine like terms:

- $ x^2 $: $ 3x^2 + 4x^2 = 7x^2 $
- $ y^2 $: $ -y^2 + 8y^2 = 7y^2 $
- $ xy $: $ -7xy + 7xy = 0 $
- $ y $: $ 4y $
- $ x $: $ -3x $

So, perimeter = $ 7x^2 + 7y^2 - 3x + 4y $

✔ Answer: $ 7x^2 + 7y^2 - 3x + 4y $

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4. What should be added to $ 10z^3 - 4z^2 $ to get $ 14z^3 + z^2 - z $?

Let the unknown expression be $ A $. Then:

$$
(10z^3 - 4z^2) + A = 14z^3 + z^2 - z
$$

Solve for $ A $:

$$
A = (14z^3 + z^2 - z) - (10z^3 - 4z^2)
$$

Distribute the minus sign:

$$
= 14z^3 + z^2 - z - 10z^3 + 4z^2
$$

Combine like terms:

- $ z^3 $: $ 14z^3 - 10z^3 = 4z^3 $
- $ z^2 $: $ z^2 + 4z^2 = 5z^2 $
- $ z $: $ -z $

So, $ A = 4z^3 + 5z^2 - z $

✔ Answer: $ 4z^3 + 5z^2 - z $

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5. Find the product.

#### a. $ (4 - 2d - d)(3 - 2d) $

Wait — first simplify: $ 4 - 2d - d = 4 - 3d $

So: $ (4 - 3d)(3 - 2d) $

Use distributive property (FOIL):

$$
= 4 \cdot 3 + 4 \cdot (-2d) + (-3d) \cdot 3 + (-3d) \cdot (-2d)
$$
$$
= 12 - 8d - 9d + 6d^2
$$
$$
= 12 - 17d + 6d^2
$$

✔ Answer: $ 6d^2 - 17d + 12 $

#### b. $ (x^2 + 2y + z)(x + y^2 + z) $

This is a trinomial × trinomial. Multiply term by term:

We’ll do it systematically:

Multiply each term in the first bracket with each in the second:

1. $ x^2(x) = x^3 $
2. $ x^2(y^2) = x^2y^2 $
3. $ x^2(z) = x^2z $

4. $ 2y(x) = 2xy $
5. $ 2y(y^2) = 2y^3 $
6. $ 2y(z) = 2yz $

7. $ z(x) = zx $
8. $ z(y^2) = zy^2 $
9. $ z(z) = z^2 $

Now write all terms:

$$
x^3 + x^2y^2 + x^2z + 2xy + 2y^3 + 2yz + zx + zy^2 + z^2
$$

Group similar terms (none are alike except possibly $ zx $ and $ xz $, but same):

Note: $ zx = xz $, so we can write as $ xz $

Final expression:

$$
x^3 + x^2y^2 + x^2z + 2xy + 2y^3 + 2yz + xz + y^2z + z^2
$$

✔ Answer:
$ x^3 + x^2y^2 + x^2z + 2xy + 2y^3 + 2yz + xz + y^2z + z^2 $

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6. Divide $ (32x^4y^3 - 16x^3y^4) $ by $ (-8x^2y) $

Divide each term separately:

First term:
$$
\frac{32x^4y^3}{-8x^2y} = \frac{32}{-8} \cdot x^{4-2} \cdot y^{3-1} = -4x^2y^2
$$

Second term:
$$
\frac{-16x^3y^4}{-8x^2y} = \frac{-16}{-8} \cdot x^{3-2} \cdot y^{4-1} = 2xy^3
$$

So, quotient = $ -4x^2y^2 + 2xy^3 $

✔ Answer: $ -4x^2y^2 + 2xy^3 $

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7. If $ 4z $ books cost ₹$ (8z^2 + 5z) $, what is the cost of one book?

Total cost = $ 8z^2 + 5z $
Number of books = $ 4z $

Cost per book = $ \frac{8z^2 + 5z}{4z} $

Split:

$$
= \frac{8z^2}{4z} + \frac{5z}{4z} = 2z + \frac{5}{4}
$$

So, cost of one book = $ 2z + \frac{5}{4} $ rupees

✔ Answer: $ 2z + \frac{5}{4} $ ₹

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8. Expand the following: $ (4x + y)^3 $

Use the identity:
$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$

Let $ a = 4x $, $ b = y $

Then:

$$
(4x)^3 + 3(4x)^2(y) + 3(4x)(y)^2 + y^3
$$

Calculate each term:

- $ (4x)^3 = 64x^3 $
- $ 3(16x^2)(y) = 48x^2y $
- $ 3(4x)(y^2) = 12xy^2 $
- $ y^3 $

So, expansion:

$$
64x^3 + 48x^2y + 12xy^2 + y^3
$$

✔ Answer: $ 64x^3 + 48x^2y + 12xy^2 + y^3 $

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✔ Final Answers Summary:

1. Degrees:
a. 1
b. 1
c. 3

2. Descending order: a, b

3. Perimeter: $ 7x^2 + 7y^2 - 3x + 4y $

4. Add: $ 4z^3 + 5z^2 - z $

5. Products:
a. $ 6d^2 - 17d + 12 $
b. $ x^3 + x^2y^2 + x^2z + 2xy + 2y^3 + 2yz + xz + y^2z + z^2 $

6. Quotient: $ -4x^2y^2 + 2xy^3 $

7. Cost per book: $ 2z + \frac{5}{4} $ ₹

8. Expansion: $ 64x^3 + 48x^2y + 12xy^2 + y^3 $

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