Chapter 1.4 Rational expressions

Stanley ocken m19500 precalculus chapter 14 rational

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The answer is a reduced fraction since numerator does not factor. Stanley Ocken M19500 Precalculus Chapter 1.4: Rational Expressions

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Welcome Algebra with numerical fractions Algebra with polynomial fractions Exercises Quiz Review Example 22: Use the LCM from Example 16 to simplify x x 2 + 3 x + 2 + x + 3 x 2 + 4 x + 4 Solution: LCD= the LCM found above = ( x + 1)( x + 2) 2 x x 2 + 3 x + 2 = x ( x + 2)( x + 1) = x ( x + 2 ) ( x + 2)( x + 1) ( x + 2 ) = x 2 + 2 x ( x + 2) 2 ( x + 1) x + 3 x 2 + 4 x + 4 = x + 3 ( x + 2) 2 = ( x + 3) ( x + 1 ) ( x + 2) 2 ( x + 1 ) = x 2 + 4 x + 3 ( x + 2) 2 ( x + 1) Add the rewritten fractions by adding their numerators: x 2 + 2 x ( x + 2) 2 ( x + 1) + x 2 + 4 x + 3 ( x + 2) 2 ( x + 1) = 2 x 2 + 6 x + 3 ( x + 2) 2 ( x + 1) Try to reduce the fraction by factoring the numerator. To see if that is possible, apply the Quadratic Factoring Criterion: 2 x 2 + 6 x + 3 = ax 2 + bx + c with a = 1 , b = 6 , and c = 3 . Then b 2 - 4 ac = 6 2 - 4 · 2 · 3 = 36 - 24 = 12 = 2 3 . Since this is not a whole number, the numerator does not factor and therefore the fraction cannot be reduced. Answer: x x 2 + 3 x + 2 + x + 3 x 2 + 4 x + 4 = 2 x 2 + 6 x + 3 ( x + 2) 2 ( x + 1) Stanley Ocken M19500 Precalculus Chapter 1.4: Rational Expressions
Welcome Algebra with numerical fractions Algebra with polynomial fractions Exercises Quiz Review Special issues with minus signs Minus signs can be annoying, especially when we deal with fractions. The nicest way to write - 0 . 7 as a fraction is - 7 10 . Also OK is - 7 10 . However, 7 - 10 is inelegant: convert it to one of othe other forms as soon as possible. When you reduce a polynomial fraction, or find the LCM of polynomials, make sure that all polynomial factors have a positive leading coefficient. The basic identity you need to know is that a - x = - ( x - a ) = - 1( x - a ) . Example 23: Reduce each of the following fractions: a) 4 - x x - 4 b) x 2 - 4 x +4 4 - x 2 Solutions: a) 4 - x x - 4 = - 1( x - 4) 1 · ( x - 4) = - 1 b) x 2 - 4 x +4 4 - x 2 = ( x - 2) 2 - ( x 2 - 4) = ( x - 2)( x - 2) - 1( x - 2)( x +2) = ( x - 2) - 1( x +2) = - x - 2 x +2 or 2 - x x +2 Stanley Ocken M19500 Precalculus Chapter 1.4: Rational Expressions

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Welcome Algebra with numerical fractions Algebra with polynomial fractions Exercises Quiz Review Complex Fractions Definition A complex fraction is an expression of the form E F in which E and/or F contains a fraction (which might be written as a negative power). These contained fractions are called nested fractions. Examples of complex fractions include 1 + x x + y b x + a x + y and z + x y xy + 1 4 Procedure To simplify a complex fraction: multiply both numerator and denominator by the LCD of all the nested fractions. Stanley Ocken M19500 Precalculus Chapter 1.4: Rational Expressions
Welcome Algebra with numerical fractions Algebra with polynomial fractions Exercises Quiz Review Example 24: Rewrite y + x - 2 z without negative exponents. Solution Method 1. Rewrite x - 2 as 1 x 2 , multiply top and bottom of y + 1 x 2 z by x 2 : y + 1 x 2 z = y + 1 x 2 x 2 zx 2 = x 2 y + 1 x 2 z Solution Method 2: Multiply top and bottom of original fraction y + x - 2 z by x 2 : y + x - 2 z = ( y + x - 2 ) x 2 zx 2 = yx 2 + x - 2 x 2 zx 2 = yx 2 + 1 x 2 z Stanley Ocken M19500 Precalculus Chapter 1.4: Rational Expressions

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Welcome Algebra with numerical fractions Algebra with polynomial fractions Exercises Quiz Review Example 25: Rewrite x - 1 + y - 1 x - 2 - y - 2 as a reduced fraction.

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