Algebra Honors (Periods 5 & 6)

Simplifying Fractions 6-1

When the numerator and the denominator of an algebraic fraction have no common factor other than 1 or -1, the fraction is said to be in simplest form. To simplify a fraction, first factor the numerator and the denominator.

Simplify:

(3a + 6)/ (3a + 3b)

3(a + 2)/3(a + b)

=(a  + 2)/(a +b)    (where a ≠ -b)

REMEMBER : YOU CANNOT DIVIDE BY ZERO. You must restrict the variables in a denominator by excluding any values that would make the denominator equal to ZERO.

So with the above example, a  CANNOT EQUAL –b

Simplify

(x²-9)/(2x+1)(3+x)

(x+3)(x-3)

(2x+1)(3+x)

since x + 3 = 3 +x

you can simplify both the numerator and denominator  to

x-3

2x+1

  (where x ≠-1/2, x≠ -3)

To see which values of the variable to exclude look at the denominator of the original fraction as well. Neither 2x + 1 nor 3 +x can be equal to zero.  Since 2x + 1 ≠ 0  x ≠ =1/2  and since 3 + x≠ 0  x ≠ -3

Simplify

2x² + x - 3

2- x- x²

First factor the numerator and the denominator, using  the skills you developed from the last chapter. If you do not see any common factors, look for opposites—as in this case

(x - 1)(2x+3)

(1- x)(2 + x)

Notice that (x-1) and (1-x) are opposites.

(1 -  x) = -( x - 1)

So change the sign on the fractions and use the opposites.

That is 

(x - 1)(2x+3)

-(x- 1)(2 + x)

and that can simplify to

(2x+3)

-(2 + x)

  or

-    (2x+3)

        x+2

 (where  x≠1, x≠-2)

Solve for x

ax - a²=bx –b²  

Collect all terms with x on one side of the equation

ax – bx = a² –b²

Factor BOTH sides of the equation

x(a –b) = (a+b)(a –b)

Divide BOTH sides of the equation by the coefficient of x ( which is    a-b)

x = a + b     ( where a≠ b)