Dividing Polynomials

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1 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 2

2 Warm Up Divide using long division Divide x + 15y 3 7a 2 ab a 2x + 5y 7a b

3 Objective Use long division to divide polynomials.

4 Essential Question How do you divide polynomials using long division?

5 Polynomial long division is a method for dividing a polynomial by another polynomial of a lower degree. It is very similar to dividing numbers.

6 Example 1: Using Long Division to Divide a Polynomial Divide using long division. ( y 2 + 2y ) (y 3) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. 2y 3 y 2 + 0y + 25 Step 2 Write division in the same way you would when dividing numbers. y 3 2y 3 y 2 + 0y + 25

7 Step 3 Divide. Example 1 Continued 2y 2 + 5y+ 15 y 3 2y 3 y 2 + 0y + 25 Notice that y times 2y 2 is 2y 3. Write 2y 2 above 2y 3. (2y 3 6y 2 ) Multiply y 3 by 2y 2. Then 5y 2 + 0y subtract. Bring down the next term. Divide 5y 2 by y. (5y 2 15y) Multiply y 3 by 5y. Then 15y + 25 subtract. Bring down the next term. Divide 15y by y. (15y 45) Multiply y 3 by 15. Then subtract. 70 Find the remainder.

8 Example 1 Continued Step 4 Write the final answer. y 2 + 2y y 3 = 2y 2 + 5y y 3

9 Check It Out! Example 1a Divide using long division. (15x 2 + 8x 12) (3x + 1) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. 15x 2 + 8x 12 Step 2 Write division in the same way you would when dividing numbers. 3x x 2 + 8x 12

10 Check It Out! Example 1a Continued Step 3 Divide. 5x + 1 3x x 2 + 8x 12 (15x 2 + 5x) 3x 12 (3x + 1) 13 Notice that 3x times 5x is 15x 2. Write 5x above 15x 2. Multiply 3x + 1 by 5x. Then subtract. Bring down the next term. Divide 3x by 3x. Multiply 3x + 1 by 1. Then subtract. Find the remainder.

11 Check It Out! Example 1a Continued Step 4 Write the final answer. 15x 2 + 8x 12 3x + 1 = 5x x + 1

12 Check It Out! Example 1b Divide using long division. (x 2 + 5x 28) (x 3) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. x 2 + 5x 28 Step 2 Write division in the same way you would when dividing numbers. x 3 x 2 + 5x 28

13 Check It Out! Example 1b Continued Step 3 Divide. x + 8 x 3 x 2 + 5x 28 (x 2 3x) 8x 28 (8x 24) 4 Notice that x times x is x 2. Write x above x 2. Multiply x 3 by x. Then subtract. Bring down the next term. Divide 8x by x. Multiply x 3 by 8. Then subtract. Find the remainder.

14 Check It Out! Example 1b Continued Step 4 Write the final answer. x 2 + 5x 28 x 3 = x x 3

15 Essential Question How do you divide polynomials using long division? Step 1: Write the dividend in standard form, including terms with a coefficient of 0. Step 2: Write division in the same way as you would when dividing numbers. Step 3: Divide Step 4: Write the final quotient.

16 Objective Use synthetic division to divide polynomials.

17 synthetic division Vocabulary

18 Essential Question How do you use synthetic division to divide by a linear binomial?

19 Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form (x a).

20

21 Example 2A: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (3x 2 + 9x 2) (x 3) Step 1 Find a. Then write the coefficients and a in the synthetic division format. a = 3 For (x 3 ), a = Write the coefficients of 3x 2 + 9x 2.

22 Example 2A Continued Step 2 Bring down the first coefficient. Then multiply and add for each column Draw a box around the remainder, 52. Step 3 Write the quotient. 52 3x x 3

23 Check Multiply (x ) 3 Example 2A Continued 52 3x x 3 3x(x ) (x ) x 3 (x 3) = 3x 2 + 9x 2

24 Example 2B: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (3x 4 x 3 + 5x 1) (x + 2) Step 1 Find a. a = 2 For (x + 2), a = 2. Step 2 Write the coefficients and a in the synthetic division format Use 0 for the coefficient of x 2.

25 Example 2B Continued Step 3 Bring down the first coefficient. Then multiply and add for each column Draw a box around the remainder, 45. Step 4 Write the quotient. 3x 3 7x x x + 2 Write the remainder over the divisor.

26 Check It Out! Example 2a Divide using synthetic division. (6x 2 5x 6) (x + 3) Step 1 Find a. a = 3 For (x + 3), a = 3. Step 2 Write the coefficients and a in the synthetic division format Write the coefficients of 6x 2 5x 6.

27 Check It Out! Example 2a Continued Step 3 Bring down the first coefficient. Then multiply and add for each column Draw a box around the remainder, 63. Step 4 Write the quotient. 6x x + 3 Write the remainder over the divisor.

28 Check It Out! Example 2b Divide using synthetic division. (x 2 3x 18) (x 6) Step 1 Find a. a = 6 For (x 6), a = 6. Step 2 Write the coefficients and a in the synthetic division format Write the coefficients of x 2 3x 18.

29 Check It Out! Example 2b Continued Step 3 Bring down the first coefficient. Then multiply and add for each column There is no remainder. Step 4 Write the quotient. x + 3

30 You can use synthetic division to evaluate polynomials. This process is called synthetic substitution. The process of synthetic substitution is exactly the same as the process of synthetic division, but the final answer is interpreted differently, as described by the Remainder Theorem.

31 Example 3A: Using Synthetic Substitution Use synthetic substitution to evaluate the polynomial for the given value. P(x) = 2x 3 + 5x 2 x + 7 for x = P(2) = Write the coefficients of the dividend. Use a = 2. Check Substitute 2 for x in P(x) = 2x 3 + 5x 2 x + 7. P(2) = 2(2) 3 + 5(2) 2 (2) + 7 P(2) = 41

32 Example 3B: Using Synthetic Substitution Use synthetic substitution to evaluate the polynomial for the given value. P(x) = 6x 4 25x 3 1 3x + 5 for x = Write the coefficients of the dividend. Use 0 for the coefficient of x 2. Use a =. 3 P( 1 ) = 7 3

33 Check It Out! Example 3a Use synthetic substitution to evaluate the polynomial for the given value. P(x) = x 3 + 3x for x = P( 3) = Write the coefficients of the dividend. Use 0 for the coefficient of x 2 Use a = 3. Check Substitute 3 for x in P(x) = x 3 + 3x P( 3) = ( 3) 3 + 3( 3) P( 3) = 4

34 Check It Out! Example 3b Use synthetic substitution to evaluate the polynomial for the given value. P(x) = 5x x + 3 for x = Write the coefficients of 1 the dividend. Use a = P( 1 ) = 5 5

35 Example 4: Geometry Application Write an expression that represents the area of the top face of a rectangular prism when the height is x + 2 and the volume of the prism is x 3 x 2 6x. The volume V is related to the area A and the height h by the equation V = A h. Rearranging for A gives A = V. h A(x) = x 3 x 2 6x Substitute. x Use synthetic division. The area of the face of the rectangular prism can be represented by A(x)= x 2 3x.

36 Check It Out! Example 4 Write an expression for the length of a rectangle with width y 9 and area y 2 14y The area A is related to the width w and the length l by the equation A = l w. l(x) = y 2 14y + 45 y 9 Substitute Use synthetic division The length of the rectangle can be represented by l(x)= y 5.

37 Essential Question How do you use synthetic division to divide by a linear binomial? Write the coefficients of the dividend. In the upper left corner, write the value of a for the divisor (x-a). Copy the first coefficient in the dividend below the horizontal bar. Multiply the first coefficient by the divisor, and write the product under the next coefficient. Add the numbers in the new column. Repeat Step 2 until additions have been completed in all columns. Draw a box around the last sum. The quotient (depressed polynomial) is represented by the numbers below the horizontal bar. The boxed number is the remainder. The others are the coefficients of the polynomial quotient, in order of decreasing degree.

38 Lesson Quiz 1. Divide by using long division. (8x 3 + 6x 2 + 7) (x + 2) 8x 2 10x Divide by using synthetic division. (x 3 3x + 5) (x + 2) x 2 2x x x Use synthetic substitution to evaluate P(x) = x 3 + 3x 2 6 for x = 5 and x = ; 4 4. Find an expression for the height of a parallelogram whose area is represented by 2x 3 x 2 20x + 3 and whose base is represented by (x + 3). 2x 2 7x + 1

39 Teacher: Why didn t you do your homework? Student: The long division took too long, and the synthetic division just wasn t real.

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