Add and Subtract Rational Expressions

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Presentation transcript:

Add and Subtract Rational Expressions Section 8.5 Add and Subtract Rational Expressions

EXAMPLE 1 Add or subtract with like denominators Perform the indicated operation. 7 4x + 3 a. 2x x + 6 – 5 b. SOLUTION 7 4x + 3 a. = 7 + 3 4x 10 4x = 5 2x = Add numerators and simplify result. 2x x + 6 5 – b. x + 6 2x – 5 = Subtract numerators.

GUIDED PRACTICE for Example 1 Perform the indicated operation and simplify. a. 7 12x – 5 = 7 – 5 12x = 2 12x = 1 6x Subtract numerators and simplify results . b. 2 3x2 + 1 = 2 + 1 3x2 = 3 3x2 = 1 x2 Add numerators and simplify results. c. 4x x–2 – x = 4x–x x–2 = 3x x–2 Subtract numerators. d. 2x2 x2+1 + 2 2x2+2 x2+1 = 2(x2+1) x2+1 = Factor numerators and simplify results . = 2

Find the least common denominator of 4x2 –16 and 6x2 –24x + 24. EXAMPLE 2 Find a least common multiple (LCM) Find the least common denominator of 4x2 –16 and 6x2 –24x + 24. SOLUTION STEP 1 Factor each polynomial. Write numerical factors as products of primes. 4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2) 6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2

LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2 EXAMPLE 2 Find a least common multiple (LCM) STEP 2 Form the LCM by writing each factor to the highest power it occurs in either polynomial. LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2

EXAMPLE 3 Add with unlike denominators Add: 9x2 7 + x 3x2 + 3x SOLUTION To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that 9x2 = 32x2 and 3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1) = 9x2(x + 1). 7 9x2 x 3x2 + 3x = + 3x(x + 1) Factor second denominator. 7 9x2 x + 1 + 3x(x + 1) x 3x LCD is 9x2(x + 1).

7x + 7 9x2(x + 1) 3x2 + = 3x2 + 7x + 7 9x2(x + 1) = Multiply. EXAMPLE 3 Add with unlike denominators 7x + 7 9x2(x + 1) 3x2 + = Multiply. 3x2 + 7x + 7 9x2(x + 1) = Add numerators.

Subtract: x + 2 2x – 2 –2x –1 x2 – 4x + 3 – x + 2 2x – 2 –2x –1 EXAMPLE 4 Subtract with unlike denominators Subtract: x + 2 2x – 2 –2x –1 x2 – 4x + 3 – SOLUTION x + 2 2x – 2 –2x –1 x2 – 4x + 3 – x + 2 2(x – 1) – 2x – 1 (x – 1)(x – 3) – = Factor denominators. x + 2 2(x – 1) = x – 3 – – 2x – 1 (x – 1)(x – 3) 2 LCD is 2(x  1)(x  3). x2 – x – 6 2(x – 1)(x – 3) – 4x – 2 – = Multiply.

x2 – x – 6 – (– 4x – 2) = 2(x – 1)(x – 3) x2 + 3x – 4 = EXAMPLE 4 Subtract with unlike denominators x2 – x – 6 – (– 4x – 2) 2(x – 1)(x – 3) = Subtract numerators. x2 + 3x – 4 2(x – 1)(x – 3) = Simplify numerator. = (x –1)(x + 4) 2(x – 1)(x – 3) Factor numerator. Divide out common factor. x + 4 2(x –3) = Simplify.