I this chapter we will learn about the concept of algebraic fractions with examples.
To understand the chapter you should have basic idea about fractions and algebraic expressions.
What are algebraic fractions ?
The fraction that contains algebraic expression on numerator or denominator or both are called algebraic fractions.
How algebraic fractions are different from fraction numbers?
Fraction number contains integers or constant values on both numerator and denominators.
While algebraic fractions contain polynomials on either numerator or denominator.
The polynomials in the algebraic expression contains both constants and variables.
Note:
Remember that in fractions the denominator cannot be equal to 0 otherwise the number would get ” Not defined “.
Examples of algebraic fractions
Given below are examples of algebraic fractions with explanation.
\mathtt{( i) \ \frac{5}{x+3}}
Numerator contains number ⟹ 5
Denominator contains polynomial ⟹ ( x +3 )
\mathtt{( ii) \ \frac{x^{2} +4x+6}{-11}}
Numerator contains polynomial ⟹ \mathtt{x^{2} +4x+6}
Denominator contains integer ⟹ -11
\mathtt{(iii) \ \frac{\mathtt{x} +13}{x^{3} +6}}
Numerator contains polynomial ⟹ (x + 13)
Denominator contains polynomial ⟹ \mathtt{\left( x^{3} +6\right)}
\mathtt{( iv) \ \ \frac{x^{4} +12x^{2} +4}{1}}
Numerator contains polynomial ⟹ \mathtt{x^{4} +12x^{2} +4}
Denominator contains number ⟹ 1
\mathtt{( v) \ \ \frac{1-x^{2}}{1+x^{4}}}
Numerator contains polynomial ⟹ \mathtt{1-x^{2}}
Denominator contains polynomial ⟹ \mathtt{1+x^{4}}
Can we have denominator 0 in algebraic fraction ?
No!!
In math, division of any number with 0 is ” not defined “. It means that we cannot get fixed value after division with 0.
So in algebraic fraction, if denominator is polynomial then its value should not be equal to 0.
For example, consider the below algebraic fraction;
\mathtt{\Longrightarrow \ \frac{5}{x-4}}
Here;
numerator contains number ⟹ 5
denominator contains polynomial ⟹ x – 4
Let value of x = 4;
Then fraction value becomes;
\mathtt{\Longrightarrow \ \frac{5}{4-4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{0}}\\\ \\ \mathtt{\Longrightarrow \ Not\ defined}
Hence, for fraction to be viable, the value of x cannot be equal to 4.
Can we break algebraic fraction into small parts ?
Off course!!
If the numerator contains a polynomial then the polynomial can be broken down into smaller components keeping the same denominator.
For example;
(i) \mathtt{\frac{\mathtt{x^{2}} +3}{\mathtt{1+x}}}
Note that the numerator contains polynomial with two terms which are separated by addition sign.
Breakdown the individual terms and keep the same denominator.
\mathtt{\Longrightarrow \ \frac{x^{2}}{1+x} +\frac{3}{1+x}}
(ii) \mathtt{\frac{\mathtt{x^{3}} +2x-5}{\mathtt{x^{2} -7}}}
The numerator is a polynomial with three terms.
The fraction can be broken down as;
\mathtt{\Longrightarrow \ \frac{x^{3}}{x^{2} -7} +\frac{2x}{x^{2} -7} -\frac{5}{x^{2} -7}}
(iii) \mathtt{\frac{10}{5\mathtt{x^{2} +11x-3}}}
Here the numerator contain single constant number 10.
Hence, the fraction cannot be broken down further.