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Simplification of Algebraic Fractions

Here we will learn simplification of algebraic fractions to its lowest term.

1. Simplify the algebraic fraction:

8a2b4a2+6ab

Solution:

8a2b4a2+6ab

We see in the given fraction the numerator is monomial and the denominator is binomial, which can be factorized.

2×2×2×a×a×b2a(2a+3b)

We can see that ‘2’ and ‘a’ are the common factors in the numerator and denominator so, we cancel the common factor ‘2’ and ‘a' from the numerator and denominator.

= 4ab(2a+3b)

2. Reduce the algebraic fraction to its lowest term:

x2+8x+12x24

Solution:

x2+8x+12x24

Each of the numerator and denominator is polynomial, which can be factorized.

= x2+6x+2x+12(x)2(2)2

 = x(x+6)+2(x+6)(x+2)(x2)

= (x+2)(x+6)(x+2)(x2)

We observed that in the numerator and denominator (x + 2) is the common factor and there is no other common factor. Now, we cancel the common factor from the numerator and denominator.

= (x+6)(x2)


3. Reduce the algebraic fraction to its lowest form:

5x245x2x12

Solution:

5x245x2x12

Each of the numerator and denominator is polynomial, which can be factorized.

= 5(x29)x24x+3x12

= 5[(x)2(3)2]x(x4)+3(x4)

= 5(x+3)(x3)(x+3)(x4)

Here, in the numerator and denominator (x + 3) is the common factor and there is no other common factor. Now, we cancel the common factor from the numerator and denominator.

= 5(x3)(x4)


4. Simplify the algebraic fraction:

x413x2+362x2+10x+12

Solution:

5x245x2x12

Each of the numerator and denominator is polynomial, which can be factorized.

= x49x24x2+362(x2+5x+6)

= x2(x29)4(x29)2(x2+2x+3x+6)

= (x24)(x29)2[x(x+2)+3(x+2)]

= (x24)(x29)2(x+2)(x+3)[Since,a2b2=(a+b)(ab)]

= (x+2)(x2)(x+3)(x3)2(x+2)(x+3)

Here, in the numerator and denominator (x + 2) and (x + 3) are the common factors and there is no other common factor. Now, we cancel the common factors from the numerator and denominator.

= (x2)(x3)(x3)2

5. Reduce the algebraic fraction to its lowest term:

x2+5x22x2+x6÷4x296x2+7x3

Solution:

x2+5x22x2+x6÷4x296x2+7x3

Each of the numerator and denominator of each fraction are polynomial, which can be factorized.

Now by factorizing each polynomial we get;

3x2 + 5x – 2 = 3x2 –x + 6x – 2

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

                 = (x + 2)(3x – 1)

2x2 + x – 6 = 2x2 - 3x - 4x - 6

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

                = (x + 2)(2x - 3)

4x2 – 9 = (2x)2 - (3)2

           = (2x + 3)(2x – 3)

6x2 + 7x – 3 = 6x2 – 2x + 9x – 3

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

                  = (2x + 3)(3x – 1)

Therefore, we have

(x+2)(3x1)(x+2)(2x3)÷(2x+3)(2x3)(2x+3)(3x1)

= (3x1)(2x3)×(2x3)(3x1)

= (3x1)2(2x3)2

= 9x26x+14x212x+9

 

6. Reduce the algebraic fraction to its lowest form:

 1x23x+2+1x25x+6+1x24x+3

Solution:

1x23x+2+1x25x+6+1x24x+3

= 1x22xx+2+1x23x2x+6+1x2x3x+3

= 1x(x2)1(x2)+1x(x3)2(x3)+1x(x1)3(x1)

= 1(x2)(x1)+1(x3)(x2)+1(x1)(x3)

= 1×(x3)(x2)(x1)(x3)+1×(x1)(x3)(x2)(x1)+1×(x2)(x1)(x3)(x2)

= (x3)(x2)(x1)(x3)+(x1)(x3)(x2)(x1)+(x2)(x1)(x3)(x2)

= (x3)+(x1)+(x2)(x1)(x2)(x3)

= (3x6)(x1)(x2)(x3)

= 3(x2)(x1)(x2)(x3)

= 3(x1)(x3)

 

7. Simplify the algebraic fraction:

3xx2+5xx24

Solution:

3xx2+5xx24

= 3xx2+5xx2(2)2

= 3xx2+5x(x+2)(x2)

= 3x×(x+2)(x2)(x+2)+5x(x+2)(x2)

= 3x(x+2)5x(x2)(x+2)

= 3x2+6x5x(x2)(x+2)

= 3x2+x(x2)(x+2)

= x(3x+1)(x2)(x+2)






8th Grade Math Practice

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