A Fraction of Polynomial Expressions. to Simplify a Rational Expres

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

polynomial A rational expression is of the form ; a fraction of polynomial expressions. polynomial To simplify a rational expression means… 1. The answer is in ‘lowest terms’; the numerator and denominator have no common factors. 2. Usually, the answer may be left factored.

Simplify each rational expression. 60 48 Ex 1:  Ex 2:  220 12 4

y2  25 12 4rr2 Ex 3:  Ex 4:  y3 125 rr32 2

Simplify each expression using addition, subtraction, multiplication, or division. 53 57 Ex 5:  Ex 6:  24 20 12 8

14 25 Ex 7:  25 24

5a22 12 a  4 25 a  20 a  4 Ex 8: Divide:  a4216 a 2 a

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

For addition and subtraction (as noted earlier), you need to find a least common denominator (LCD) and re-write both rational expressions with that LCD before adding or subtracting. Here are a few examples of finding an LCD.

Find the least common denominator of each problem. 2 12 1 Ex 9:  n23 n  10 n 2  4 n  4 n 2  5 n

4 x Ex 10:  x36 x 2  9 x 2 x 2  8 x  6

Add or Subtract: 3mm 5 40 Ex 11:    m2 m  2 m2  4

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

4x 8 2 Ex 12:    3x 4 3 x2 4 x x

2xx 6 5 7 Ex 13:    x226 x  9 x  9 x  3

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

Many mathematicians do not consider an expression left with a radical in its denominator to be simplified. Clearing a radical from a denominator is called rationalizing the denominator. Examine the following example.

Ex 14: 2 2 7  7 7 7 27  49 2 7 2  or 7 77

53 Ex 15: Rationalize the denominator:  28

(It is easier to rationalize the denominator if the radicals are simplified first.)

Ex 16: Rationalize the denominator. Assume the variable is positive.

2  18x3

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

4 Ex 17: Rationalize the denominator.  3 9

Conjugates are expressions of the form a b and a b . When rationalizing expressions with a sum or difference in the denominator (with square roots), the numerator and denominator must be multiplied by the conjugate of the denominator. Why do we do this? Because the product of conjugates is always a rational number or expression (no roots). Examine the following. (2 11)(2  11)  4  2 11  2 11  121 4 121 4 11 7 4 3x 4  3 x  16  12 x  12 x  9 x2    16 9 x2 16 9x ( 3 6)( 3  6)  9  18  18  36 36 3 When multiplying conjugates with square roots, the square root is always eliminated!

When rationalizing a denominator with a binomial denominator with a square root, multiply numerator and denominator by the conjugate of the denominator. See the example 18 below. Ex 18: t5 ( t  5) ( t  5)  t5 ( t  5) ( t  5) t2 5 t 5 t 25  In the denominator, the 'inner' and 'outer' products were eliminated. t 2  25 tt10 25  t  25

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

Ex 19: Rationalize the denominator. 43  43

Ex 20: Rationalize the denominator and simplify. 81x2  16  32x 

Occasionally in calculus there is a need to rationalize the numerator. Ex 21: Rationalize the numerator in this expression. Simplify. 23x   43x2 

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

A complex fraction or complex rational expression is a quotient in which the numerator and/or the denominator is(are) a fractional expression or expressions.

To simplify a complex rational expression, combine the numerator and/or the denominator into a single quotient (one fraction). Convert to a division problem and perform the division.

Ex 22: Simplify the following expression.

33  ()x h22 x  h

Ex 23: Simplify the following.

77  x h 22 x   h

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

xa  Ex 24: Simplify: xa33 xa

Ex 25: Simplify this expression. 33  2xa 1 2 1  xa

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

ba  Ex 26: Simplify this complex expression. ab 11  ab

Ex 27: Simplify: 1  3 x  2  4  x x

Ex 28: Simplify: xy  yx  xy22  yx22

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

Ex 29: Simplify: 52x  xx13 x 7  xx13

(x h )33  5( x  h )  ( x  5 x ) Ex 30: Simplify:  h

Express each as a quotient. (Examples 10 – 11) Ex 31: xx4  Ex 32: xx3 

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

For this next type of example, usually the answer should be left in factored form. 27  Ex 33: xx33

These next types of examples will occur often in calculus when finding some types of derivatives. Ex 34: Factor the expression below. (6x 5)3 (2)( x 2  4)(2 x )  ( x 2  4) 2 (3)(6 x  5) 2 (6) 

Ex 35: Factor the expression below. 4x2 ( x 1) 2 ( x  2)  3 x ( x  1)( x  2) 2 

MA 15800 Lesson 4 Notes Summer 2016 Simplifying Algebraic Expression (part 2)

Ex 36: Factor the expression below. (x2 2) 3 (2 x )  x 2 (3)( x 2  2) 2 (2 x )  232 (x  2)

Ex 37: Express as a quotient: xx3/2 1/2

Ex 38: Simplify the expression. (6x 5)3 (2)( x 2  4)(2 x )  ( x 2  4) 2 (3)(6 x  5) 2 (6) 