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10-6 Dividing Polynomials Preview Warm Up California Standards Lesson Presentation 10-6 Dividing Polynomials Warm Up Divide. 1. m2n ÷ mn4 2. 2x3y2 ÷ 6xy 3. (3a + 6a2) ÷ 3a2b Factor each expression. 4. 5x2 + 16x + 12 5. 16p2 – 72p + 81 10-6 Dividing Polynomials California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, by using these techniques. 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 10-6 Dividing Polynomials To divide a polynomial by a monomial, you can first write the division as a rational expression. Then divide each term in the polynomial by the monomial. 10-6 Dividing Polynomials Additional Example 1: Dividing a Polynomial by a Monomial Divide (5x3 – 20x2 + 30x) ÷ 5x Write as a rational expression. Divide each term in the polynomial by the monomial 5x. Divide out common factors. x2 – 4x + 6 Simplify. 10-6 Dividing Polynomials Check It Out! Example 1a Divide. (8p3 – 4p2 + 12p) ÷ (–4p2) Write as a rational expression. Divide each term in the polynomial by the monomial –4p2. Divide out common factors. Simplify. 10-6 Dividing Polynomials Check It Out! Example 1b Divide. (6x3 + 2x – 15) ÷ 6x Write as a rational expression. Divide each term in the polynomial by the monomial 6x. Divide out common factors in each term. Simplify. 10-6 Dividing Polynomials Division of a polynomial by a binomial is similar to division of whole numbers. 10-6 Dividing Polynomials Additional Example 2A: Dividing a Polynomial by a Binomial Divide. Factor the numerator. Divide out common factors. x+5 Simplify. 10-6 Dividing Polynomials Additional Example 2B: Dividing a Polynomial by a Binomial Divide. Factor both the numerator and denominator. Divide out common factors. Simplify. 10-6 Dividing Polynomials Helpful Hint Put each term of the numerator over the denominator only when the denominator is a monomial. If the denominator is a polynomial, try to factor first. 10-6 Dividing Polynomials Check It Out! Example 2a Divide. Factor the numerator. Divide out common factors. k+5 Simplify. 10-6 Dividing Polynomials Check It Out! Example 2b Divide. Factor the numerator. Divide out common factors. b–7 Simplify. 10-6 Dividing Polynomials Check It Out! Example 2c Divide. Factor the numerator. Divide out common factors. s+6 Simplify. 10-6 Dividing Polynomials Recall how you used long division to divide whole numbers as shown at right. You can also use long division to divide polynomials. An example is shown below. (x2 + 3x + 2) ÷ (x + 2) Divisor x+1 x + 2) x2 + 3x + 2 x2 + 2x x+2 x+2 0 Quotient Dividend 10-6 Dividing Polynomials Using Long Division to Divide a Polynomial by a Binomial Step 1 Write the binomial and polynomial in standard form. Step 2 Divide the first term of the dividend by the first term of the divisor. This the first term of the quotient. Step 3 Multiply this first term of the quotient by the binomial divisor and place the product under the dividend, aligning like terms. Step 4 Subtract the product from the dividend. Step 5 Bring down the next term in the dividend. Step 6 Repeat Steps 2-5 as necessary until you get 0 or until the degree of the remainder is less than the degree of the binomial. 10-6 Dividing Polynomials Additional Example 3A: Polynomial Long Division Divide using long division. Check your answer. (x2 +10x + 21) ÷ (x + 3) Write in long division Step 1 x + 3 ) x2 + 10x + 21 form with expressions in standard form. x Step 2 x + 3 ) x2 + 10x + 21 Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. 10-6 Dividing Polynomials Additional Example 3A Continued Divide using long division. (x2 +10x + 21) ÷ (x + 3) x Step 3 x + 3 ) x2 + 10x + 21 x2 + 3x x Step 4 x + 3 ) x2 + 10x + 21 –(x2 + 3x) 0 + 7x Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. Subtract the product from the dividend. 10-6 Dividing Polynomials Additional Example 3A Continued Divide using long division. x Step 5 x + 3 ) x2 + 10x + 21 –(x2 + 3x) 7x + 21 Bring down the next term in the dividend. x +7 Step 6 x + 3 ) x2 + 10x + 21 Repeat Steps 2-5 as necessary. 2 –(x + 3x) 7x + 21 –(7x + 21) The remainder is 0. 0 10-6 Dividing Polynomials Additional Example 3A Continued Check: Multiply the answer and the divisor. (x + 3)(x + 7) x2 + 7x + 3x + 21 x2 + 10x + 21 10-6 Dividing Polynomials Helpful Hint When the remainder is 0, you can check your simplified answer by multiplying it by the divisor. You should get the numerator. 10-6 Dividing Polynomials Additional Example 3B: Polynomial Long Division Divide using long division. x – 4) x2 – 2x – 8 Write in long division form. x+ 2 x – 4) x2 – 2x – 8 –(x2 – 4x) 2x – 8 –(2x – 8) x2 ÷ x = x Multiply x (x – 4). Subtract. 0 Bring down the 8. 2x ÷ x = 2. Multiply 2(x – 4). Subtract. The remainder is 0. 10-6 Dividing Polynomials Additional Example 3B Continued Check: Multiply the answer and the divisor. (x + 2)(x – 4) x2 – 4x + 2x – 8 x2 – 2x + 8 10-6 Dividing Polynomials Check It Out! Example 3a Divide using long division. (2y2 – 5y – 3) ÷ (y – 3) Step 1 y – 3) Step 2 2y y – 3) 2y2 – 5y – 3 2y2 – 5y – 3 Write in long division form with expressions in standard form. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. 10-6 Dividing Polynomials Check It Out! Example 3a Continued Divide using long division. (2y2 – 5y – 3) ÷ (y – 3) Step 3 2y y – 3) 2y2 – 5y – 3 2y2 – 6y 2y Step 4 y – 3) 2y2 – 5y – 3 –(2y2 – 6y) 0+ y Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. Subtract the product from the dividend. 10-6 Dividing Polynomials Check It Out! Example 3a Continued Divide using long division. 2y Bring down the next term Step 5 y – 3) 2y2 – 5y – 3 in the dividend. –(2y2 – 6y) y –3 Step 6 2y + 1 y – 3) 2y2 – 5y – 3 –(2y2 – 6y) y–3 –(y – 3) 0 Repeat Steps 2–5 as necessary. The remainder is 0. 10-6 Dividing Polynomials Check It Out! Example 3a Continued Check: Multiply the answer and the divisor. (y – 3)(2y + 1) 2y2 + y – 6y – 3 2y2 – 5y – 3 10-6 Dividing Polynomials Check It Out! Example 3b Divide using long division. (a2 – 8a + 12) ÷ (a – 6) a – 6) a2 – 8a + 12 a– 2 a – 6) a2 – 8a + 12 –(a2 – 6a) –2a + 12 –(–2a + 12) 0 Write in long division form. a2 ÷ a = a Multiply a (a – 6). Subtract. Bring down the 12. –2a ÷ a = –2. Multiply –2(a – 6). Subtract. The remainder is 0. 10-6 Dividing Polynomials Check It Out! Example 3b Continued Check: Multiply the answer and the divisor. (a – 6)(a – 2) a2 – 2a – 6a + 12 a2 – 8a + 12 10-6 Dividing Polynomials Sometimes the divisor is not a factor of the dividend, so the remainder is not 0. Then the remainder can be written as a rational expression. 10-6 Dividing Polynomials Additional Example 4: Long Division with a Remainder Divide (3x2 + 19x + 26) ÷ (x + 5) x + 5) 3x2 + 19x + 26 3x + 4 x + 5) 3x2 + 19x + 26 –(3x2 + 15x) 4x + 26 –(4x + 20) 6 Write in long division form. 3x2 ÷ x = 3x. Multiply 3x(x + 5). Subtract. Bring down the 26. 4x ÷ x = 4. Multiply 4(x + 5). Subtract. The remainder is 6. Write the remainder as a rational expression using the divisor as the denominator. 10-6 Dividing Polynomials Additional Example 4 Continued Divide (3x2 + 19x + 26) ÷ (x + 5) Write the quotient with the remainder. 10-6 Dividing Polynomials Check It Out! Example 4a Divide. m + 3) 3m2 + 4m – 2 Write in long division form. 3m – 5 m + 3) 3m2 + 4m – 2 –(3m2 + 9m) 3m2 ÷ m = 3m. Multiply 3m(m + 3). Subtract. Bring down the –2. –5m ÷ m = –5 . –5m – 2 –(–5m – 15) Multiply –5(m + 3). Subtract. 13 The remainder is 13. 10-6 Dividing Polynomials Check It Out! Example 4a Continued Divide. Write the remainder as a rational expression using the divisor as the denominator. 10-6 Dividing Polynomials Divide. Check It Out! Example 4b y – 3) y2 + 3y + 2 y+ 6 y – 3) y2 + 3y + 2 –(y2 – 3y) 6y + 2 –(6y –18) 20 y+6+ Write in long division form. y2 ÷ y = y. Multiply y(y – 3). Subtract. Bring down the 2. 6y ÷ y = 6. Multiply 6(y – 3). Subtract. The remainder is 20. Write the quotient with the remainder. 10-6 Dividing Polynomials Sometimes you need to write a placeholder for a term using a zero coefficient. This is best seen if you write the polynomials in standard form. 10-6 Dividing Polynomials Additional Example 5: Dividing Polynomials That Have a Zero Coefficient Divide (x3 – 7 – 4x) ÷ (x – 3). (x3 – 4x – 7) ÷ (x – 3) x – 3) x3 + 0x2 – 4x – 7 x2 x – 3) x3 + 0x2 – 4x – 7 –(x3 – 3x2) 3x2 – 4x Write the polynomials in standard form. Write in long division form. Use 0x2 as a placeholder for the x2 term. x3 ÷ x = x2 Multiply x2(x – 3). Subtract. Bring down –4x. 10-6 Dividing Polynomials Additional Example 5 Continued x2 + 3x + 5 x – 3) x3 + 0x2 – 4x – 7 –(x3 – 3x2) 3x2 – 4x –(3x2 – 9x) 5x – 7 –(5x – 15) 8 (x3 – 4x – 7) ÷ (x – 3) = 3x3 ÷ x = 3x Multiply x2(x – 3). Subtract. Bring down –4x. Multiply 3x(x – 3). Subtract. Bring down – 7. Multiply 5(x – 3). Subtract. The remainder is 8. 10-6 Dividing Polynomials Remember! Recall from Chapter 7 that a polynomial in one variable is written in standard form when the degrees of the terms go from greatest to least. 10-6 Dividing Polynomials Check It Out! Example 5a Divide (1 – 4x2 + x3) ÷ (x – 2). (x3 – 4x2 + 1) ÷ (x – 2) x – 2) x3 – 4x2 + 0x + 1 x2 – 2x – 4 x – 2) x3 – 4x2 + 0x + 1 –(x3 – 2x2) – 2x2 + 0x –(–2x2 + 4x) – 4x + 1 –(–4x + 8) –7 Write in standard form. Write in long division form. Use 0x as a placeholder for the x term. x3 ÷ x = x2 Multiply x2(x – 2). Subtract. Bring down 0x. – 2x2 ÷ x = –2x. Multiply –2x(x – 2). Subtract. Bring down 1. Multiply –4(x – 2). Subtract. 10-6 Dividing Polynomials Check It Out! Example 5a Continued Divide (1 – 4x2 + x3) ÷ (x – 2). (1 – 4x2 + x3) ÷ (x – 2) = 10-6 Dividing Polynomials Check It Out! Example 5b Divide (4p – 1 + 2p3) ÷ (p + 1). (2p3 + 4p – 1) ÷ (p + 1) p + 1) 2p3 + 0p2 + 4p – 1 2p2 – 2p + 6 p + 1) 2p3 + 0p2 + 4p – 1 –(2p3 + 2p2) – 2p2 + 4p –(–2p2 – 2p) 6p – 1 –(6p + 6) –7 Write in standard form. Write in long division form. Use 0p2 as a placeholder for the p2 term. p3 ÷ p = p2 Multiply 2p2(p + 1). Subtract. Bring down 4p. –2p2 ÷ p = –2p. Multiply –2p(p + 1). Subtract. Bring down –1. Multiply 6(p + 1). Subtract. 10-6 Dividing Polynomials Check It Out! Example 5b Continued (2p3 + 4p – 1) ÷ (p + 1) = 10-6 Dividing Polynomials Lesson Quiz: Part I Add or Subtract. Simplify your answer. 1. (12x2 – 4x2 + 20x) ÷ 4x 2. 2x + 3 3. x–2 4. x+3 3x2 – x + 5 10-6 Dividing Polynomials Lesson Quiz: Part II Divide using long division. 5. (x2 + 4x + 7) (x + 1) 6. (8x2 + 2x3 + 7) (x + 3)