Adding and Subtracting Algebraic Fractions

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A addition and subtraction of algebraic fractions is done similarly to adding and subtracting numerical fractions, the difference is that in algebraic fractions we deal with polynomials.

When the denominators of algebraic fractions are the same, just add or subtract the numerators and keep the denominator.

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However, if the denominators are different, we must write equivalent fractions with equal denominators to then do the addition or subtraction. In this case, calculate the MMC of polynomials.

Algebraic fractions with like denominators

If the denominators of algebraic fractions are the same, we add or subtract the numerators and keep the denominator.

Examples:

a) Calculate $\dpi{120} \mathrm{\frac{7x}{y^2}+\frac{3x}{y^2} }$ .

$\dpi{120} \mathrm{\frac{7x}{y^2}+\frac{3x}{y^2} \frac{7x+3x}{y^2} \frac{10x}{y^2 } }$

b) Calculate $\dpi{120} \mathrm{\frac{9 + a}{b-1}-\frac{a-b}{b-1} }$ .

$\dpi{120} \mathrm{\frac{9 + a}{b-1}-\frac{a-b}{b-1} \frac{9 + a - (a-b)}{b-1} \frac{ 9 -b}{b-1} }$

Algebraic fractions with different denominators

If the denominators of the algebraic fractions are different, we calculate the LCM of the denominators and write equivalent fractions with the same denominator.

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Then we calculate the addition or subtraction just like in the previous case, of equal denominators.

Examples:

a) Calculate $\dpi{120} \mathrm{\frac{x}{2y}+\frac{y}{2x}}$ .

We factor each of the polynomials that are in the denominator:

$\dpi{120} \mathrm{2y 2\cdot y}$

$\dpi{120} \mathrm{2x 2\cdot x}$

The MMC is the product between the factors, but without repeating the same factors:

$\dpi{120} \mathrm{\Rightarrow MMC 2\cdot y\cdot x 2yx}$

Note that we do not repeat the number 2, which appears in the factorization of the two polynomials.

Using MMC, we rewrite equivalent fractions with the same denominator:

$\dpi{120} \mathrm{\frac{x}{2y}+\frac{y}{2x} \frac{x^2}{2yx}+ \frac{y^2}{2yx}}$

Finally, we calculate the sum of algebraic fractions that already have the same denominator:

$\dpi{120} \mathrm{\frac{x}{2y}+\frac{y}{2x} \frac{x^2+y^2}{2yx}}$

b) Calculate $\dpi{120} \mathrm{\frac{2a}{a^2-9} - \frac{7}{a+3}}$ .

To find the MMC between the polynomials that are in the denominator, we factor each one of them.

$\dpi{120} \mathrm{a^2 - 9 a^2 - 3^2 (a-3)\cdot (a+3)}$ → factoring the difference of two squares

$\dpi{120} \mathrm{a+ 3 a+3}$ → stays the same

The MMC is the product between the factors, but without repeating the same factors.

$\dpi{120} \mathrm{\Rightarrow MMC (a+3)\cdot (a-3)}$

Note that we do not repeat (a + 3), which appears in the factorization of the two polynomials.

Using MMC, we rewrite equivalent fractions with the same denominator:

$\dpi{120} \mathrm{\frac{2a}{a^2-9} - \frac{7}{a+3} \frac{2a}{(a+3)\cdot (a-3)} -\frac{7.(a-3)}{(a+3)\cdot (a-3)}}$

Finally, we calculate the sum of algebraic fractions that already have the same denominator:

$\dpi{120} \mathrm{\frac{2a}{a^2-9} - \frac{7}{a+3} \frac{2a - 7(a-3)}{(a+3)\ cdot (a-3)} \frac{2a-7a+21}{(a+3)\cdot (a-3)} \frac{-5a+21}{(a+3)\cdot (a-3) )} }$

You may also be interested:

Multiplication of polynomials
Division of polynomials - Key method
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List of Least Common Multiple Exercises – MMC

Adding and Subtracting Algebraic Fractions

Algebraic fractions with like denominators

Algebraic fractions with different denominators

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