Problems on Algebraic Fractions

Here we will learn how to simplify the problems on algebraic fractions to its lowest term.

1. Reduce the algebraic fractions to their lowest terms: \(\frac{x^{2}  -  y^{2}}{x^{3}  -  x^{2}y}\)

Solution:

\(\frac{x^{2}  -  y^{2}}{x^{3}  -  x^{2}y}\)

Factorizing the numerator and denominator separately and cancelling the common factors we get,

= \(\frac{(x  +  y) (x  -  y)}{x^{2} (x  -  y)} \)

= \(\frac{x  +  y}{x^{2}}\)

2. Reduce to lowest terms \(\frac{x^{2}  +  x  -  6}{x^{2}  -  4}\)

Solution:

\(\frac{x^{2}  +  x  -  6}{x^{2} -  4}\)

Step 1: Factorize the numerator x\(^{2}\) + x – 6

                                         = x\(^{2}\) + 3x – 2x – 6

                                         = x(x + 3) – 2(x + 3)

                                         = (x + 3) (x – 2)

Step 2: Factorize the denominator: x\(^{2}\) – 4

                                             = x\(^{2}\) – 2\(^{2}\)

                                             = (x + 2) (x – 2)

Step 3: From steps 1 and 2: \(\frac{x^{2}  +  x  -  6}{x^{2}  -  4}\)

                                     = \(\frac{x^{2}  +  x  -  6}{x^{2}  -  2^{2}}\)

                                     = \(\frac{(x  +  3) (x  -  2)}{(x  +  2) (x  -  2)}\)

                                     = \(\frac{(x  +  3)}{(x  +  2)}\)

3. Simplify the algebraic fractions \(\frac{36x^{2}  -  4}{9x^{2}  +  6x  +  1}\)

Solution:

\(\frac{36x^{2}  -  4}{9x^{2}  +  6x  +  1}\)

Step 1: Factorize the numerator: 36x\(^{2}\) – 4

                                           = 4(9x\(^{2}\) – 1)

                                           = 4[(3x)\(^{2}\) – (1)\(^{2}\)]

                                           = 4(3x + 1) (3x – 1)

Step 2: Factorize the denominator: 9x\(^{2}\) + 6x + 1

                                             = 9x\(^{2}\) + 3x + 3x + 1

                                             = 3x(3x + 1) + 1(3x + 1)

                                             = (3x + 1) (3x + 1)

Step 3: Simplification of the given expression after factorizing the numerator and the denominator:

\(\frac{36x^{2}  -  4}{9x^{2}  +  6x  +  1}\)

= \(\frac{4(3x  +  1)(3x  -  1)}{(3x  +  1)(3x  +  1)}\)

= \(\frac{4(3x  -  1)}{(3x  +  1)}\)

4. Reduce and simplify: \(\frac{8x^{3}y^{2}z}{2xy^{3}} of \left ( \frac{5x^{5}y^{2}z^{2}}{25xy^{3}z} \div \frac{7xy^{2}}{35x^{2}yz^{3}}\right )\)

Solution:

\(\frac{8x^{3}y^{2}z}{2xy^{3}} of \left ( \frac{5x^{5}y^{2}z^{2}}{25xy^{3}z} \div \frac{7xy^{2}}{35x^{2}yz^{3}}\right )\)

= \(\frac{8x^{3}y^{2}z}{2xy^{3}} of \frac{5x^{5}y^{2}z^{2}}{25xy^{3}z} \times \frac{35x^{2}yz^{3}}{7xy^{2}}\)

= \(\frac{4x^{3}y^{2}z}{xy^{3}} \left ( \frac{x^{5}y^{2}z^{2}}{xy^{3}z} \times \frac{x^{2}yz^{3}}{xy^{2}} \right )\)

= 4x\(^{10 - 3}\) ∙ y\(^{-3}\) ∙ z\(^{5}\)

= \(\frac{4x^{7}\cdot z^{5}}{y^{3}}\)

5. Simplify: \(\frac{2x^{2}  -  3x  -  2}{x^{2}  +  x  -  2} \div \frac{2x^{2}  +  3x  +  1}{3x^{2}  +  3x  -  6}\)

Solution:

\(\frac{2x^{2}  -  3x  -  2}{x^{2}  +  x   -   2} \div \frac{2x^{2}  +  3x  +  1}{3x^{2}  +  3x  -  6}\)

Step 1: First factorize each of the polynomials separately:

2x\(^{2}\) – 3x – 2 = 2x\(^{2}\) – 4x + x – 2

                 = 2x(x – 2) + 1 (x – 2)

                 = (x – 2) (2x + 1)

x\(^{2}\) + x – 2 = x\(^{2}\) + 2x - x – 2

              = x(x + 2) - 1 (x + 2)

              = (x + 2) (x - 1)

2x\(^{2}\) + 3x + 1 = 2x\(^{2}\) + 2x + x + 1

                 = 2x(x + 1) + 1 (x + 1)

                 = (x + 1) (2x + 1)

3x\(^{2}\) + 3x – 6 = 3[x\(^{2}\) + x – 2]

                 = 3[x\(^{2}\) + 2x - x – 2]

                 = 3[x(x + 2) – 1(x + 2)]                   

                 = 3[(x + 2) (x - 1)]

                 = 3[(x + 2) (x - 1)]

                 = 3(x + 2) (x - 1)

Step 2: Simplify the given expressions by substituting with their factors

\(\frac{2x^{2}  -  3x  -  2}{x^{2}  +  x  -  2} \div \frac{2x^{2}  +  3x  +  1}{3x^{2}  +  3x  -  6}\)

= \(\frac{2x^{2}  -  3x  -  2}{x^{2}  +  x  -  2} \times \frac{3x^{2}  +  3x  -  6}{2x^{2}  +  3x  +  1}\)

= \(\frac{(x  -  2) (2x  +  1)}{(x  +  2) (x  -  1)}\times\frac{3(x  +  2) (x  -  1)}{(x  +  1) (2x  +  1)}\)

= \(\frac{3(x  -  2)}{(x  +  1)}\)

8th Grade Math Practice

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