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3.7 Warm Up Solve the equation. 1. √(x + 3) + 8 = 15 2. √(8 – 3x) + 5 = 6 3. 4√(2x + 1) – 7 = 1

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3.7 Divide Polynomials

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To divide a polynomial by a monomial Write each term as a fraction Divide each term separately Simplify

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SOLUTION Divide a polynomial by a monomial EXAMPLE 1 Divide 4x 3 + 8x 2 + 10x by 2x. Method 1 : Write the division as a fraction. Write as fraction. Divide each term by 2x. = 2x 2 + 4x + 5 8x28x2 4x34x3 2x2x 2x2x 10x 2x2x ++= Simplify. (4x 3 + 8x 2 + 10x) 2x 4x 3 + 8x 2 +10x 2x2x =

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GUIDED PRACTICE for Example 1 (6x 3 + 3x 2 –12x) 3x 1. 2x 2 + x – 4 ANSWER (12y 4 – 16y 3 + 20y 2 ) 4y 2. ANSWER 3y 3 – 4y 2 + 5y

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Divide polynomial by a binomial Divide the 1 st term of the polynomial by the 1 st term of the binomial (step 1) Multiply the whole binomial by step 1 Subtract from polynomial as in long division Continue this pattern Remainder will be placed over the binomial

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SOLUTION EXAMPLE 2 Divide a polynomial by a binomial Divide x 2 + 2x – 3 by x – 1. STEP 1 Divide the first term of x 2 + 2x – 3 by the first term of x – 1. Multiply x and x – 1. x x – 1x 2 + 2x – 3 x 2 – x 3x3x Subtract x 2 – x from x 2 + 2x. Think : x 2 x = ?

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EXAMPLE 3 Divide a polynomial by a binomial Divide 2x 2 + 11x – 9 by 2x – 3. x 2x 2 –3x 2x – 3 2x 2 + 11x – 9 14x – 9 12 Multiply x and 2x – 3. Subtract 2x 2 – 3x. Bring down – 9. Multiply 7 and 2x – 3. Subtract 14x – 21. ANSWER (2x 2 + 11x – 9) (2x – 3) = x + 7 + 12 2x – 3 14x – 21 + 7

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GUIDED PRACTICE for Examples 2 and 3 3. Divide: (a 2 + 3a – 4) (a + 1) a + 2 + ANSWER – 6 a + 1 4. Divide: (9b 2 + 6b + 8) (3b – 4) ANSWER 3b + 6 + 32 3b – 4

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EXAMPLE 4 Rewrite polynomials Divide 5y + y 2 + 4 by 2 + y. Rewrite polynomials. Multiply y and y + 2. y 2 + 2y Subtract y 2 + 2y. Bring down 4. 3y + 4 Multiply 3 and y + 2. 3y + 6 Subtract 3y + 6. – 2 y + 2 y 2 + 5y + 4 y ANSWER (5y + y 2 + 4) (2 + y) = y + 3 + y + 2 – 2 + 3

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EXAMPLE 5 Insert missing terms Divide 13 + 4m 2 by – 1 + 2m. Rewrite polynomials, Insert missing term. 2m2m 2m – 1 4m 2 + 0m + 13 Multiply 2m and 2m – 1. 4m 2 – 2m Subtract 4m 2 – 2m. Bring down 13. 2m + 13 Multiply 1 and 2m – 1. 2m – 1 Subtract 2m – 1. 14 ANSWER (13 + 4m 2 ) (– 1 + 2m) = 2m + 1 + 2m – 1 14 + 1

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GUIDED PRACTICE for Examples 4, 5, and 6 5. Divide: (8m – 7 + 4m 2 ) (5 + 2m) ANSWER 2m – 1+ 2m + 5 – 2 6. Divide: (n 2 – 6) (– 3 + n) n + 3 + ANSWER n – 3 3

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EXAMPLE 6 Rewrite and graph a rational function Graph y = 2x – 1 x – 2 SOLUTION STEP 1 + x – h a Rewrite the rational function in the form y = k.k. x – 22x – 1 2x – 4 3 2 So, y = + 2. x – 2 3

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EXAMPLE 6 Rewrite and graph a rational function STEP 2 Graph the function.

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Graph y = 3x + 1 x + 1 GUIDED PRACTICE for Examples 4, 5, and 6

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