Rational Algebraic Expressions

Factoring Polynomials

1

x²+10x+16

(x+2)(x+8)

2

3x²-5x-12

= (3x-4)(x+3)

3

x²-6x+8

= (x-2)(x-4)

4

2x²-x-6

= (2x+5)(x-2)

5

x²-36

= (x+6)(x-6)

Simplify Rational Expressions

1

2x+4/ x²+5x+6

= 2(x+2)/ (x+3)(x+2)
= cancel the both x+2
= 2/ (x+3)

2

4x³-6x²/ 2x

= 2x²(2x-3)/2x

= x(2x-3)

3

x²-5x+6/ x-3

= (x-2)(x-3)/ (x-3)
= cancel both (x-3)
= x-2

4

4x³/ 8x²

= (1/2)x^3-2
= (1/2)x
= x/2

5

(3x+9)/ (3x+15)

= (3x+9/ (3x+15)
=3(x+3)/ 3(x/5)
=(x+30/ (x+5)

Adding Rational Expressions

1

2/x + 5/x

= 7/x

2

2/3+ 5/3

= 7/3

3

8/x + 7/x

= 15/x

4

x/ 2x+3 + 5/2x+3

= = x+5/2x+3

5

3x/ 5+3 + x/1+4

= 3x/ 6+7

Subtracting Rational Expressions

1

5y-3/ x - 3y+2/x

= 5y-3-(3y+2)

= 5y-3-3y-2/ x
= 2-5/ x

2

5y/ x+4 - y-1/ x+4

= 5y-(y-1)/ x+4
= 5y-y+1/ x+4
= 4y+2/ x+4

3

6x-2/ x - 2x-5/x

= 6x-2-(2x-5)/ x
= 6x-2-2x+5/ x
= 4x+3/ x

4

9z-7/x - 2z-3/ x

= 9z-7-(2z-3)/ x
= 9z-7-2z+3/ x
= 7z-4/ x

5

2-4b/ b - 1-3b/ b

= 2-4b-(1-3b)/ b
= 2-4b-1+3b/ b
= 1-b/ b

Dividing Rational Expressions

1

3x⁴/ 4 ÷  9x/10

= 5x³/ 6

2

3x/2 ÷  6y/x

=3x/2 × x/6y
= x/2 ×  x/2y
= x²/4y

3

7p/ p+2 ÷ p+2/p

= 7p/ p+2 × p/ p+2
= 7p2/ (p+2)²

4

5(a+2)/ b ÷  a+2/ b

= 5 (a+2)/b × b/ (a+2)
= 5

5

5x³/3xy ÷ 3x²/ 24xy

= 5x3/ 3xy × 24xy/ 3x²
= 5x × 8/3
= 40/3 x