Exercise 1

Simplify:

1 \frac {x ^ 2 - 3x}{x ^2 + 3x}

 

2 \frac {x ^ 2 - 3x}{3 - x}

 

3 \frac {x ^ 2 - 5x + 6}{x ^2 + 7x + 12}

 

4 \frac {x ^ 2 - 2x - 3}{x ^2 - x - 2}

 

5 \frac {x ^ 3 - 19x - 30}{x^3 - 3x ^2 - 10x}

 

Exercise 2

Calculate:

\frac {1}{x + 1} + \frac {2x}{x ^2 - 1} - \frac{1}{x - 1} =

Exercise 3

Calculate:

\frac {x + 2}{x ^ 3 - 1} - \frac {1}{x - 1} =

Exercise 4

Calculate:

1 \frac {x ^2 - 2x} {x ^2 - 5x + 6} \cdot \frac {x ^2 + 4x + 4}{x ^2 - 4}

 

2 \frac{9 - 6x + x^2}{9 - x^2} \cdot \frac{x ^2 - 5x + 6}{3x^2 - 9x}

Exercise 5

Calculate:

1 \frac{x + 2}{x ^2 + 4x + 4} \div \frac{x ^2 - 4}{x ^3 + 8}

 

2 \frac{x ^3 + 3x^2 - 4x - 12}{x ^ 2 + 2x - 3} \div \frac {4x - 2x^2}{x ^3 - 2x^2 + x}

Exercise 6

Calculate:

( x + \frac {x}{x - 1} ) \cdot ( x - \frac{x}{x - 1})

Exercise 7

Calculate:

( x + \frac {x}{x - 1} ) \div ( x - \frac{x}{x - 1})

 

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 1

Simplify:

1 \frac {x ^2 - 3x} {x ^2 + 3x}

 

x is the factor both in the numerator and the denominator:

 

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

 

Cancel x from the numerator and the denominator:

 

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

 

2 \frac {x ^2 - 3x}{ 3 - x}

 

- x is the common factor in the numerator and the denominator:

 

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

 

Make it -x:

 

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

 

Cancel x - 3 from both the numerator and denominator. The resultant answer will be - x.

 

3. \frac {x ^2 + x - 2}{x ^3 - x^2 - x + 1}

 

Factorize the numerator and the denominator:

 

= \frac { (x - 1) \cdot (x + 2)}{(x - 1) \cdot (x ^2 - 1)}

 

x - 1 will be cancelled from both numerator and the denominator:

 

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

 

3 \frac {x ^ 2 - 5x + 6}{x ^2 + 7x + 12}

 

Factorize the numerator and the denominator:

 

=\frac { (x - 2) \cdot (x - 3)}{ (x - 3) \cdot (x - 4)} =

 

Cancel x - 3 from both numerator and the denominator:

 

=\frac {(x - 2)}{(x - 4)}

 

4    \frac {x ^ 2 - 2x - 3}{x ^2 - x - 2}

 

Factorize the numerator and the denominator:

 

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

 

Cancel x + 1 from the numerator and the denominator:

 

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

 

5 \frac {x ^ 3 - 19x - 30}{x^3 - 3x ^2 - 10x}

 

Factorize the numerator and the denominator:

 

= \frac {(x + 2) \cdot (x + 3) \cdot (x - 5)}{x \cdot (x + 2) \cdot (x - 5)}

 

The common factors will be cancelled:

 

= \frac {x + 3}{x}

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 2

Calculate:

\frac {1}{x + 1} + \frac {2x}{x ^2 - 1} - \frac{1}{x - 1} =

 

Take L.C.D of the above expression:

 

\frac { x - 1 + 2x - x - 1}{x ^2 - 1}

 

\frac {2x - 2}{x ^2 - 1}

 

x ^2 - 1 can be written as (x - 1) (x + 1):

 

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

 

(x - 1) will be cancelled from both the numerator and the denominator. Hence, the resultant answer will be:

 

= \frac {2}{x + 1}

 

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 3

Calculate:

=\frac {x + 2}{x ^ 3 - 1} - \frac {1}{x - 1} =

 

x^3 - 1 = (x - 1) \cdot (x ^2 + x + 1)

 

Least Common Denominator ( L.C.D) is (x - 1) \cdot (x ^2 + x + 1)

 

=\frac {x + 2 - (x ^2 + x + 1)}{(x - 1) \cdot (x ^2 + x + 1)}

 

The parentheses in the numerator will be opened and the signs of all the terms inside it will be reversed:

 

=\frac { x + 2 - x^2 - x - 1}{(x - 1) \cdot (x ^2 + x + 1)}

 

= \frac { - (x ^2 - 1)}{ (x - 1) \cdot (x ^2 + x + 1)}

 

x ^2 - 1 can be factored as (x - 1) (x + 1)

 

= \frac {- (x - 1) (x + 1)}{(x - 1) \cdot (x ^2 + x + 1)}

 

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

 

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 4

Calculate:

1    \frac {x ^2 - 2x} {x ^2 - 5x + 6} \cdot \frac {x ^2 + 4x + 4}{x ^2 - 4}

 

\frac {(x ^2 - 2x) \cdot (x ^2 + 4x + 4)}{(x ^2 - 5x + 6) \cdot (x ^2 - 4)}

 

\frac { x (x - 2) \cdot (x + 2) ^2} {(x - 2) \cdot (x - 3) \cdot (x - 2) \cdot (x + 2)}

 

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

 

2   \frac{9 - 6x + x^2}{9 - x^2} \cdot \frac{x ^2 - 5x + 6}{3x^2 - 9x}

 

= \frac { ( 9 - 6x + x ^2) \cdot (x ^2 - 5x + 6)}{( 9 - x^2) \cdot (3x ^2 - 9x )}

 

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

 

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

 

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 5

Calculate:

1   \frac{x + 2}{x ^2 + 4x + 4} \div \frac{x ^2 - 4}{x ^3 + 8}

 

The second expression will be reversed because of the division sign:

 

= \frac {(x + 2) \cdot (x ^3 + 8)}{x ^2 + 4x + 4) \cdot (x ^2 - 4)}

 

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

 

= \frac {x ^2 - 2x + 4}{x ^2 - 4}

2   \frac{x ^3 + 3x^2 - 4x - 12}{x ^ 2 + 2x - 3} \div \frac {4x - 2x^2}{x ^3 - 2x^2 + x}

 

The second fraction will be reversed because of the division sign:

 

= \frac {(x ^ 3 + 3x ^2 - 4x - 12) \cdot (x ^3 - 2x ^2 + x)}{(x ^2 +2x - 3) \cdot (4x - 2x ^2)}

 

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

 

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

 

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

 

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 6

Calculate:

( x + \frac {x}{x - 1} ) \cdot ( x - \frac{x}{x - 1})

Since, (x + 1) (x - 1) = (x - 1) ^2, so ( x + \frac {x}{x - 1} ) \cdot ( x - \frac{x}{x - 1}) can be written as:

 

x ^2 - (\frac {x}{x - 1}) ^2

 

= x ^2 - \frac {x ^2}{(x - 1) ^2 }

 

Take L.C.D of both the expressions:

 

= \frac { x ^2 \cdot (x - 1) ^ 2 - x ^2 } { (x - 1) ^ 2}

 

= \frac { x^2 [ (x - 1) ^2 - 1]}{(x - 1) ^2 }

 

= \frac { x ^2 \cdot (x - 1 - 1) \cdot (x - 1 + 1)}{(x - 1) ^2 }

 

= \frac {x ^2 \cdot (x - 2) \cdot x }{(x - 1) ^2 }

 

= \frac { x^3 \cdot (x - 2) }{(x - 1) ^2 }

 

Solution of exercise Algebraic Fractions Worksheet

Solution of exercise 7

Calculate:

( x + \frac {x}{x - 1} ) \div ( x - \frac{x}{x - 1})

 

Take L.C.D of the second expression:

 

= \frac { x \cdot (x - 1) + x}{ x - 1} \div \frac {x \cdot (x - 1) - x}{x - 1}

 

\frac { x ^2 - x + x}{x - 1} \div \frac {x ^2 - x - x}{x - 1}

 

\frac {x ^2 }{x - 1} \div \frac {x ^2 - 2x }{x - 1}

 

\frac {x ^2 \cdot (x - 1)}{x \cdot (x - 2) \cdot (x - 1)}

 

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

 

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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