Simplifying Rational Expressions

Question 1

Factorise 28x^2 - 2x - 6

Answer 1

• ( First take out what’s common )
• 2 ( 14x^2 - x - 3 )
• ( Find the x values of the equation using your calculator and rearrange them )
• x = 1 / 2
• x = - 3 / 7
• 2x - 1 = 0
• 7x + 3 = 0
• 2 ( 2x - 1 )( 7x + 3 )

Question 2

Simplify 6x^2 - 5x - 21 / ( 2x + 3 )

Answer 2

• ( First factorise the numerator )
• ( 2x + 3 )( 3x - 7 ) / ( 2x + 3 )
• ( Now we can cancel them out )
• = 3x - 7

Question 3

Simplify 12x^2 + 26x - 16 / 6x^2 - 11x - 72

Answer 3

• ( Factorise the numerator and denominator )
• 2 ( 2x - 1 )( 3x + 8 ) / ( 3x + 8 )( 2x - 9 )
• = ( 4x - 2 )( 3x + 8 ) / ( 3x + 8 )( 2x - 9 )
• ( Cancel common factors )
• ( 4x - 2 ) / ( 2x - 9 )
• = 2 ( x - 1 ) / ( 2x - 9 )

Question 4

Given that 6x^3 + 3x^2 - 84x / 6x^2 - 33x + 42 = ax ( x + b ) / x + c
Find the values of a,b and c

Answer 4

• ( Factorise the numerator )
• 3x ( 2x^2 + x - 28 )
• ( Divide the denominator by 3 )
• 2x^2 - 11x + 14
• 3x ( 2x^2 + x - 28 ) / 2x^2 - 11x + 14
• ( Factorise numerator and denominator )
• 3x ( 2x - 7 )( x + 4 ) / ( 2x - 7 )( x - 2 )
• ( Cancel common factors )
• 3x ( x + 4 ) / ( x - 2 )
• a = 3
• b = 4
• c = - 2

Question 5

A rectangle has a length of ( 1 + root 3 )cm and area of root 80cm^2
Find the width of the rectangle in cm.
Express your answer in the form a root 5 + b root c. where a, b and c are integers.

Answer 5

• ( width = area / length )
• root 80 / 1 + root 3
• ( Rationalise the denominator )
• ( root 80 / 1 + root 3 ) x ( 1 - root 3 / 1 - root 3 )
• root 80 - root 240 / 1 - 3 = 4 root 5 - 4 root 15 / - 2
• • 2 root 5 + 2 root 15
• a = - 2
• b = 2
• c = 15

Question 6

*Jack and Jill walked up the hill at 3km / h and returned down the hill at 4km / h.
The entire trip took 5 hours and 10 minutes, including a half hour rest break.

How long was the trail