Skip to content# Algebraic Fractions Simplifying Worksheet and Examples

## What is an Algebraic Fraction?

## Steps to Simplify Algebraic Fractions

**Solution:**

## Steps for Adding and Subtracting Algebraic Fractions

## Steps for Multiplying Algebraic Fractions

## Steps for Dividing Algebraic Fractions

**Solution:**

## Solving Equations with Algebraic Fractions

**Solution:**

## Conclusion

## Worksheet on Algebraic Fraction

**Algebraic Fractions**

- Algebraic fractions are expressions that contain one or more variables in the numerator or denominator, or both.
- Algebraic fractions are similar to numerical fractions, but instead of numbers, they have algebraic expressions.

In this article, we will discuss:

**What is an Algebraic Fraction?****Steps for Adding, Subtracting, Multiplying and Dividing Algebraic Fractions**

Here is one more link to practice a few extra questions: Maths Genie Algebraic Fractions Questions

- An algebraic fraction is a fraction whose numerator or denominator or both are algebraic expressions.

- For example, (3x
^{2}+ 2x)/(x + 1) is an algebraic fraction.

- To simplify an algebraic fraction, follow the following steps:

**Step #1: Factorise the numerator and denominator**

**Step #2: Cancel out any common factors in the numerator and denominator**

**Step #3: Write the simplified fraction**

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**Solved Example: **

**Question 1: Simplify (3x ^{2} + 9x)/ (x^{2} + 4x)**

Solution:

**Step #1:**Factorise the numerator and denominator

**3x (x + 3)/x(x + 4)**

**Step #2:**Cancel out the common factor ‘x’:

**3(x + 3)/(x + 4)**

**Step #3:**Write the simplified fraction

**3(x + 3)/(x + 4)**

**Practice Questions**

Question 1: Simplify (2y^{2} + 6y) / (y^{2} + 3y)

**Step #1:** Factorize the numerator and denominator:

**2y(y + 3) / y(y + 3)**

**Step #2:** Cancel out the common factor 'y + 3':

**2y / y**

**= 2**

**Step #3:** Write the simplified fraction: 2

Question 2: Simplify (4a^{2} - 16) / (a^{2} - 4)

**Solution:**

**Step #1:** Factorize the numerator and denominator:

**4(a + 4)(a - 4) / (a + 2)(a - 2)**

**Step #2:** Cancel out the common factor 'a + 4' and 'a - 4':

**4 / (a + 2)**

**Step #3:** Write the simplified fraction:

**4 / (a + 2)**

- To add or subtract algebraic fractions, follow the following steps:

**Step #1: Find a common denominator**

**Step #2: Rewrite each fraction with the common denominator**

**Step #3: Add or subtract the numerators**

**Step #4: Simplify the resulting fraction, if necessary**

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**Solved Example: **

**Question 1: Add (2x + 1)/(x ^{2} – 1) and (x – 1)/(x + 1)**

Solution:

**Step #1:**Find a common denominator

**(x ^{2} – 1)(x + 1)**

**Step #2:**Rewrite each fraction with the common denominator

**Step #3:**Add or subtract the numerators

**Step #4:**Simplify the resulting fraction

- To multiply algebraic fractions, follow the following steps:

**For multiplication:**

**Step #1: Multiply the numerators**

**Step #2: Multiply the denominators**

**Step #3: Simplify the resulting fraction, if necessary**

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**Solved Example**

**Question: Multiply (2x + 1)/ (x ^{2} – 1) and (x – 1)/(x + 1)**

Solution:

**Step #1:**Multiply the numerators

**(2x + 1)(x – 1)**

**Step #2:**Multiply the denominators

**(x ^{2} – 1)(x + 1)**

**Step #3:**Simplify the resulting fraction

**(2x ^{2} – x – 1)/(x^{3} – x)**

- To multiply algebraic fractions, follow the following steps:

**For division:**

**Step #1: Flip the second fraction (divisor) and turn it into its reciprocal**

**Step #2: Multiply the first fraction (dividend) with the reciprocal of the second fraction (divisor)**

**Step #3: Simplify the resulting fraction, if necessary**

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**Solved Example**

**Question: Divide (2x + 1)/(x ^{2} – 4x + 3) by (x – 1)/(x – 3)**

Solution:

**Step #1:**Flip the second fraction to get

**(x – 3)/(x – 1)**

**Step #2:**Multiply the first fraction with the reciprocal of the second fraction to get

**Step #3:**Simplify the resulting fraction by factoring the quadratic in the denominator and cancelling any common factors. The numerator can also be simplified using the distributive property.

The simplified expression is (2x + 1)/(x – 1)^{2}

**Practice Questions**

Question 1: Simplify the algebraic fraction: (3x^{2} - 12x) / (x^{2} - 4)

**Step #1:** Factorize the numerator and denominator:

**(3x ^{2} - 12x) / (x^{2} - 4) **

**= 3x(x - 4) / (x + 2)(x - 2)**

**Step #2:** The simplified form of the algebraic fraction is 3x(x - 4) / (x + 2)(x - 2).

Question 2: Simplify the algebraic fraction: (2x^{2} + 5x - 3) / (x^{2} - 4x + 4)

**Solution:**

**Step #1: **Factorize the numerator and denominator.

**(2x ^{2} + 5x - 3) / (x^{2} - 4x + 4) **

**= (2x - 1)(x + 3) / (x - 2) ^{2}**

**Step #2: **The simplified form of the algebraic fraction is (2x - 1)(x + 3) / (x - 2)^{2}.

When solving equations with algebraic fractions,

- The first step is to get rid of the fractions by finding a common denominator.
- Then, the equation can be solved using standard techniques like simplifying and factoring.

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**Solved Example**

**Question: Solve for x: (2x + 1)/(x – 2) + 1 = 3/(x – 2)**

Solution:

**Step #1:**Find the common denominator by multiplying both sides by (x – 2), to get

**(2x + 1) + (x – 2) = 3**

**Step #2:**Simplify the resulting equation by combining like terms, to get

**3x – 1 = 3**

**Step #3:**Solve for x by adding 1 to both sides and then dividing by 3, to get

**x = 4/3**

**Practice Questions**

Question 1: Simplify the algebraic fraction: (5a^{2} + 7ab) / (3a^{2} - 4ab)

**Step #1: **Factorize the numerator and denominator.

**(5a ^{2} + 7ab) / (3a^{2} - 4ab) **

**= a(5a + 7b) / b(3a - 4b)**

**Step #2: **The simplified form of the algebraic fraction is a(5a + 7b) / b(3a - 4b).

Question 2: Simplify the algebraic fraction: (2x^{3} - 6x^{2} + 4x) / (4x^{2} - 8x + 4)

**Solution:**

**Step #1: **Factorize the numerator and denominator.

**(2x ^{3} - 6x^{2} + 4x) / (4x^{2} - 8x + 4) **

**= 2x(x - 1)(x - 2) / 2(x - 1)(x - 2)**

**Step #2: **The simplified form of the algebraic fraction is x(x - 1) / (x - 1).

- Algebraic fractions is a keystone used for solving several problems at the basic algebra level.
- What make algebraic fractions work out is that, you should be well aware of the rules for simplifying, adding, subtracting, multiplying, dividing, solving equations, and everything else within the scope of the subject. By having the knack and moving on the fractions can be simply conquered.

**Question 1:** Simplify the algebraic fraction:

**(3a ^{3} - 9a^{2}b + 6ab^{2}) / (6a^{2}b - 12ab^{2} + 6b^{3})**

**Question 2:** Simplify the algebraic fraction:

**(x ^{4} - 4x^{2} + 4) / (x^{2} - 2x + 1)**

**Question 3:** Simplify the algebraic fraction:

**(3xy ^{2} - 6x^{2}y) / (2x^{2}y - 4xy^{2})**

**Question 4:** Simplify the algebraic fraction:

**(4x ^{3} - 8x^{2} + 4x) / (2x^{2} - 4x + 2)**