A fraction represents equal parts of a collection or whole. Algebraic fraction means a fraction whose numerator or denominator is a polynomial expression. Check the examples of algebraic fractions, its definition, and solved example questions. You can also see how to perform addition, multiplication, subtraction, and division between algebraic fractions.
Algebraic Fraction Definition
Fractions that have a polynomial expression in the numerator and denominator are called algebraic fractions. Denominators of the algebraic fractions can never be zero. You can write every polynomial as an algebraic fraction with a denominator.
Examples of Algebraic Fractions
- (x² + 2x + 3) / 3 is an algebraic fraction with integral denominator 3.
- (x + 3) / 5 is an algebraic fraction with integral denominator 5.
- 6 / (a + 5b + 3) is an algebraic fraction with integral numerator 6.
- 2 / (x + 3y) is an algebraic fraction with integral numerator 2.
- (x + y) / (x² + 10x + 7) is an algebraic fraction with numerator as linear polynomial and denominator as a quadratic polynomial.
- (y² – 11y + 32) / (y + 2) is an algebraic fraction with numerator as quadratic polynomial and denominator as a linear polynomial.
Operations with Algebraic Fractions
Below mentioned are the various operations that can be performed on algebraic fractions.
1. Reducing Fraction
To reduce the algebraic fraction, first, find the factors of numerator and denominator. Later cancel the common factors.
Example: (12x³y²) / (28xy)
(12x³y²) / (28xy) = (4 * 3 * x * x * x * y * y) / (7 * 4 * x * y)
= (3 * x * x * y) / 7 = (3x²y) / 7 = (3/7) x²y
2. Multiplying Algebraic Fractions
While multiplying 2 fractions, get the factors of the fraction and then reduce it to the lowest terms. Then multiply the numerators together, and denominators together to check the result.
Example: [(x + 1) / (5y + 10)] * [(y+2) / (x² + 2x + 1)]
= [(x + 1) / ((5(y + 2))] * [(y+2) / (x² + x + x + 1)]
= [(x + 1) / (5] * [1 / (x (x + 1) + 1 (x + 1)] = [(x + 1) / (5] * [1 / ((x + 1) (x + 1)]
= (1/5) * (1 / (x + 1)) = 1 / (5(x + 1))
3. Adding Fractions
You must have a common denominator to add two fractions. Simply, get the lcm or multiply those two denominators. Then, add the numerators.
Example: [(x – 4) / (x + 1)] + [3 / (x + 2)]
= [(x – 4) (x + 2) / (x + 1) (x + 2)] + [(3(x + 1)) / (x + 1) (x + 2)]
= [(x² – 4x + 2x – 8) / (x + 1) (x + 2)] + [(3x + 3) / (x + 1) (x + 2)]
= [(x² – 2x – 8) / (x + 1) (x + 2)] + [(3x + 3) / (x + 1) (x + 2)] = [(x² – 2x – 8 + 3x + 3) / (x + 1) (x + 2)]
= (x² + x – 5) / [(x + 1) (x + 2)]
4. Subtracting fractions
To subtract fractions, make a common denominator by finding the LCD and changing each fraction to an equivalent fraction. Then subtract those fractions.
Example: (2 / x) – (3 / y)
= (2y / xy) – (3x / xy) = (2y – 3x) / xy
5. Dividing Fractions
To divide algebraic fractions, invert the second fraction and multiply.
Example: (2 / x²) / (5 / x)
= (2 / x²) * (x / 5) = 2x / 5x² = 2 / 5x
Example Questions
Example 1.
Solve [5 / (x + 6)] + [10 / (x + 1)]
Solution:
Given fraction is [5 / (x + 6)] + [10 / (x + 1)]
Both fractions denominators are different. So find the lcm
LCM of (x + 6) and (x + 1) is (x + 6) * (x + 1)
[5 / (x + 6)] + [10 / (x + 1)] = [(5(x + 1)) / (x + 6) (x + 1)] + [(10(x + 6)) / (x + 6) (x + 1)]
= [(5x + 5) / (x + 6) (x + 1)] + [(10x + 60) / (x + 6) (x + 1)]
= [(5x + 5 + 10x + 60) / (x + 6) (x + 1)] = [(15x + 65) / (x + 6) (x + 1)]
[5 / (x + 6)] + [10 / (x + 1)] = [5(3x + 13) / (x + 6) (x + 1)]
Example 2.
Solve [(x² + 13x + 35) / (x + 4)] / [(x² – 3x – 40) / (x – 6)]
Solution:
Given fraction is [(x² + 13x + 35) / (x + 4)] / [(x² – 3x – 40) / (x – 6)]
[(x² + 13x + 35) / (x + 4)] / [(x² – 3x – 40) / (x – 6)] = [(x² + 13x + 35) / (x + 4)] * [(x – 6) / (x² – 3x – 40)]
= [(x² + 7x + 5x + 35) / (x + 4)] * [(x – 6) / (x² – 8x + 5x – 40)] = [(x(x + 7) + 5 (x + 7) / (x + 4)] * [(x – 6) / (x(x – 8) + 5(x – 8)]
= [((x + 7) (x + 5) / (x + 4)] * [(x – 6) / ((x – 8) (x + 5)]
= [(x + 7) (x + 5) (x – 6)] / [(x + 4) (x – 8) (x + 5)] = [(x + 7) (x – 6)] / [(x + 4) (x – 8)]
Example 3.
Perform the indicated operation.
(i) [10 / y] – [15 / y² – 10y + 25]
(ii) [x / (x² + 5x + 6)] – [2 / (x² + 3x + 2)]
Solution:
(i) Given fraction is [10 / y] – [15 / y² – 10y + 25]
= [10 / y] – [15 / (y² – 5y – 5y + 25]
= [10 / y] – [15 / (y(y – 5) – 5(y – 5)] = [10 / y] – [15 / (y – 5)(y – 5)]
= [10 (y – 5)² / y(y – 5)²] – [15y / y(y – 5)²]
= (10 (y – 5)² – 15y) / y(y – 5)² = 10(y² – 10y + 25) / y(y – 5)²
= [10(y² + 25) – 100y – 15y] / [y(y – 5)²] = [10(y² + 25) – 115y] / [y(y – 5)²]
(ii) Given fraction is [x / (x² + 5x + 6)] – [2 / (x² + 3x + 2)]
= [x / (x² + 3x + 2x + 6)] – [2 / (x² + 2x + x + 2)]
= [x / (x(x + 3) + 2(x + 3)] – [2 / (x(x + 2) + 1(x + 2)]
= [x / ((x + 3)(x + 2)] – [2 / (x + 2)(x + 1)] = [x(x + 1) / (x + 2)(x + 1)(x + 3)] – [2(x + 3) / (x + 2)(x + 1)(x + 3)]
= [x² + x – 2x – 6] / [(x + 2)(x + 1)(x + 3)] = [x² – x – 6] / [(x + 2)(x + 1)(x + 3)]
= [(x – 3) (x + 2)] / [(x + 2)(x + 1)(x + 3)]
= [x – 3] / [(x + 1)(x + 3)]
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