12-6 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1

Warm Up Divide. 1. m2n ÷ mn4 2. 2x3y2 ÷ 6xy 3. (3a + 6a2) ÷ 3a2b Factor each expression. 4. 5x2 + 16x + 12 5. 16p2 – 72p + 81

Objective Divide a polynomial by a monomial or binomial.

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.

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.

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.

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. Simplify.

Division of a polynomial by a binomial is similar to division of whole numbers.

Example 2A: Dividing a Polynomial by a Binomial Divide. Factor the numerator. Divide out common factors. x + 5 Simplify.

Example 2B: Dividing a Polynomial by a Binomial Divide. Factor both the numerator and denominator. Divide out common factors. Simplify.

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. Helpful Hint

Check It Out! Example 2a Divide. Factor the numerator. Divide out common factors. Simplify. k + 5

Check It Out! Example 2b Divide. Factor the numerator. Divide out common factors. b – 7 Simplify.

Check It Out! Example 2c Divide. Factor the numerator. Divide out common factors. s + 6 Simplify.

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 Quotient ) x2 + 3x + 2 x + 1 x2 + 2x x + 2 Dividend

Example 3A: Polynomial Long Division Divide using long division. (x2 +10x + 21) ÷ (x + 3) Write in long division form with expressions in standard form. x2 + 10x + 21 ) Step 1 x + 3 x2 + 10x + 21 ) Step 2 x + 3 x Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

) ) Example 3A Continued Divide using long division. (x2 +10x + 21) ÷ (x + 3) x2 + 10x + 21 ) Step 3 x + 3 x x2 + 3x Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. x2 + 10x + 21 ) Step 4 x + 3 –(x2 + 3x) x 0 + 7x Subtract the product from the dividend.

) ) Example 3A Continued Divide using long division. x Step 5 x + 3 Bring down the next term in the dividend. x2 + 10x + 21 –(x2 + 3x) 7x + 21 x2 + 10x + 21 ) Step 6 x + 3 –(x2 + 3x) x + 7 7x + 21 –(7x + 21) Repeat Steps 2-5 as necessary. The remainder is 0.

 Example 3A Continued Check: Multiply the answer and the divisor. (x + 3)(x + 7) x2 + 3x + 7x + 21  x2 + 10x + 21

Example 3B: Polynomial Long Division Divide using long division. x2 – 2x – 8 ) x – 4 Write in long division form. x + 2 x2 ÷ x = x x2 – 2x – 8 ) x – 4 Multiply x  (x – 4 ). Subtract. –(x2 – 4x) Bring down the 8. 2x ÷ x =2. 2x – 8 –(2x – 8) Multiply 2(x – 4). Subtract. The remainder is 0.

 Example 3B Continued Check: Multiply the answer and the divisor. (x + 2)(x – 4) x2 + 2x – 4x – 8 x2 – 2x + 8 

) ) Check It Out! Example 3a Divide using long division. (2y2 – 5y – 3) ÷ (y – 3) Write in long division form with expressions in standard form. 2y2 – 5y – 3 ) Step 1 y – 3 2y 2y2 – 5y – 3 ) Step 2 y – 3 Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

Check It Out! Example 3a Continued Divide using long division. (2y2 – 5y – 3) ÷ (y – 3) 2y Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. ) Step 3 y – 3 2y2 – 5y – 3 2y2 – 6y 2y2 – 5y – 3 ) Step 4 y – 3 2y Subtract the product from the dividend. –(2y2 – 6y) 0 + y

Check It Out! Example 3a Continued Divide using long division. 2y ) Step 5 y – 3 2y2 – 5y – 3 Bring down the next term in the dividend. –(2y2 – 6y) y – 3 2y + 1 Repeat Steps 2–5 as necessary. 2y2 – 5y – 3 ) Step 6 y – 3 –(2y2 – 6y) –(y – 3) The remainder is 0.

Check It Out! Example 3a Continued Check: Multiply the answer and the divisor. (y – 3)(2y + 1) 2y2 + y – 6y – 3 2y2 – 5y – 3 

) ) Check It Out! Example 3b Divide using long division. (a2 – 8a + 12) ÷ (a – 6) a2 – 8a + 12 ) a – 6 Write in long division form. a – 2 a2 ÷ a = a a2 – 8a + 12 ) a – 6 Multiply a  (a – 6 ). Subtract. –(a2 – 6a) Bring down the 12. –2a ÷ a = –2. –2a + 12 –(–2a + 12) Multiply –2(a – 6). Subtract. The remainder is 0.

Check It Out! Example 3b Continued Check: Multiply the answer and the divisor. (a – 6)(a – 2) a2 – 2a – 6a + 12 a2 – 8a + 12 

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.

Example 4: Long Division with a Remainder Divide (3x2 + 19x + 26) ÷ (x + 5) 3x2 + 19x + 26 ) x + 5 Write in long division form. 3x + 4 3x2 ÷ x = 3x. 3x2 + 19x + 26 ) x + 5 Multiply 3x(x + 5). Subtract. –(3x2 + 15x) Bring down the 26. 4x ÷ x = 4. 4x + 26 –(4x + 20) Multiply 4(x + 5). Subtract. 6 The remainder is 6. Write the remainder as a rational expression using the divisor as the denominator.

Example 4 Continued Divide (3x2 + 19x + 26) ÷ (x + 5) Write the quotient with the remainder.

) ) Check It Out! Example 4a Divide. 3m2 + 4m – 2 m + 3 Write in long division form. 3m – 5 3m2 ÷ m = 3m. 3m2 + 4m – 2 ) m + 3 Multiply 3m(m + 3). Subtract. –(3m2 + 9m) Bring down the –2. –5m ÷ m = –5 . –5m – 2 13 –(–5m – 15) Multiply –5(m + 3). Subtract. The remainder is 13.

Check It Out! Example 4a Continued Divide. Write the remainder as a rational expression using the divisor as the denominator.

) ) Check It Out! Example 4b Divide. y2 + 3y + 2 y – 3 Write in long division form. y + 6 y2 ÷ y = y. y2 + 3y + 2 ) y – 3 Multiply y(y – 3). Subtract. –(y2 – 3y) Bring down the 2. 6y ÷ y = 6. 6y + 2 –(6y –18) Multiply 6(y – 3). Subtract. 20 The remainder is 20. y + 6 + Write the quotient with the remainder.

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.

Example 5: Dividing Polynomials That Have a Zero Coefficient Divide (x3 – 7 – 4x) ÷ (x – 3). (x3 – 4x – 7) ÷ (x – 3) Write in standard format. Write in long division form. Use 0x2 as a placeholder for the x2 term. x3 + 0x2 – 4x – 7 ) x – 3

Example 5: Dividing Polynomials That Have a Zero Coefficient Divide (x3 – 7 – 4x) ÷ (x – 3). (x3 – 4x – 7) ÷ (x – 3) Write the polynomials in standard form. x3 + 0x2 – 4x – 7 ) x – 3 Write in long division form. Use 0x2 as a placeholder for the x2 term. x2 x3 + 0x2 – 4x – 7 ) x – 3 x3 ÷ x = x2 –(x3 – 3x2) 3x2 Multiply x2(x – 3). Subtract. – 4x Bring down –4x.

) Example 5 Continued x2 + 3x + 5 x3 + 0x2 – 4x – 7 x – 3 3x3 ÷ x = 3x Multiply x2(x – 3). Subtract. – 4x Bring down – 4x. –(3x2 – 9x) Multiply 3x(x – 3). Subtract. 5x – 7 Bring down – 7. –(5x – 15) Multiply 5(x – 3). Subtract. 8 The remainder is 8. (x3 – 4x – 7) ÷ (x – 3) =

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. Remember!

) ) Check It Out! Example 5a Divide (1 – 4x2 + x3) ÷ (x – 2). Write in standard format. Write in long division form. Use 0x as a placeholder for the x term. x3 – 4x2 + 0x + 1 x – 2 ) x2 – 2x – 4 x3 ÷ x = x2 x3 – 4x2 + 0x + 1 x – 2 ) –(x3 – 2x2) – 2x2 Multiply x2(x – 2). Subtract. + 0x Bring down 0x. – 2x2 ÷ x = –2x. –(–2x2 + 4x) – 4x Multiply –2x(x – 2). Subtract. + 1 Bring down 1. –(–4x + 8) –7 Multiply –4(x – 2). Subtract.

Check It Out! Example 5a Continued Divide (1 – 4x2 + x3) ÷ (x – 2). (1 – 4x2 + x3) ÷ (x – 2) =

) ) Check It Out! Example 5b Divide (4p – 1 + 2p3) ÷ (p + 1). Write in standard format. 2p3 + 0p2 + 4p – 1 p + 1 ) Write in long division form. Use 0p2 as a placeholder for the p2 term. 2p2 – 2p + 6 p3 ÷ p = p2 2p3 – 0p2 + 4p – 1 p + 1 ) –(2p3 + 2p2) Multiply 2p2(p + 1). Subtract. – 2p2 + 4p Bring down 4p. – 2p2 ÷ p = –2p. –(–2p2 – 2p) Multiply –2p(p + 1). Subtract. 6p –1 Bring down –1. –(6p + 6) –7 Multiply 6(p + 1). Subtract.

Check It Out! Example 5b Continued (2p3 + 4p – 1) ÷ (p + 1) =

Lesson Quiz: Part I Add or Subtract. Simplify your answer. 3x2 – x + 5 1. (12x2 – 4x2 + 20x) ÷ 4x) 2. 2x + 3 3. x – 2 4. x + 3

Lesson Quiz: Part II Divide using long division. 5. (x2 + 4x + 7)  (x + 1) 6. (8x2 + 2x3 + 7)  (x + 3)