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Holt McDougal Algebra 2
6-3 Dividing Polynomials
Warm UpDivide using long division.1. 12.18 ÷ 2.1
2.
3.
2x + 5y
5.8
7a – b
Divide.
6x – 15y3
7a2 – aba
Holt McDougal Algebra 2
6-3 Dividing Polynomials
Use long division and synthetic division to divide polynomials.
Objective
Holt McDougal Algebra 2
6-3 Dividing PolynomialsPolynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.
Holt McDougal Algebra 2
6-3 Dividing Polynomials
Divide using long division.
Example 1: Using Long Division to Divide a Polynomial
(–y2 + 2y3 + 25) ÷ (y – 3)
2y3 – y2 + 0y + 25
Step 1 Write the dividend in standard form, includingterms with a coefficient of 0.
Step 2 Write division in the same way you would when dividing numbers.
y – 3 2y3 – y2 + 0y + 25
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 1a
Divide using long division. (15x2 + 8x – 12) ÷ (3x + 1)
15x2 + 8x – 12
Step 1 Write the dividend in standard form, includingterms with a coefficient of 0.
Step 2 Write division in the same way you would when dividing numbers.
3x + 1 15x2 + 8x – 12
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 1b
Divide using long division. (x2 + 5x – 28) ÷ (x – 3)
x2 + 5x – 28
Step 1 Write the dividend in standard form, includingterms with a coefficient of 0.
Step 2 Write division in the same way you would when dividing numbers.
x – 3 x2 + 5x – 28
Holt McDougal Algebra 2
6-3 Dividing Polynomials
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).
Holt McDougal Algebra 2
6-3 Dividing Polynomials
Holt McDougal Algebra 2
6-3 Dividing Polynomials
Divide using synthetic division.
Example 2A: Using Synthetic Division to Divide by a Linear Binomial
(3x2 + 9x – 2) ÷ (x – )
Step 1 Find a. Then write the coefficients and a in the synthetic division format.
Write the coefficients of 3x2 + 9x – 2.
13
For (x – ), a = .13
13
13a =
13
3 9 –2
Holt McDougal Algebra 2
6-3 Dividing PolynomialsExample 2A Continued
Step 2 Bring down the first coefficient. Then multiply and add for each column.
Draw a box around the remainder, 1 .13
13
3 9 –2 1
3
Step 3 Write the quotient.
3x + 10 +1 1
313x –
10 131
133
Holt McDougal Algebra 2
6-3 Dividing PolynomialsExample 2A Continued
3x + 10 +1 1
313x –
Check Multiply (x – ) 13
= 3x2 + 9x – 2
(x – ) 13(x – ) 1
3 (x – ) 133x + 10 +
1 1313x –
Holt McDougal Algebra 2
6-3 Dividing Polynomials
Divide using synthetic division.(3x4 – x3 + 5x – 1) ÷ (x + 2)
Step 1 Find a.
Use 0 for the coefficient of x2.
For (x + 2), a = –2.a = –2
Example 2B: Using Synthetic Division to Divide by a Linear Binomial
3 – 1 0 5 –1 –2
Step 2 Write the coefficients and a in the synthetic division format.
Holt McDougal Algebra 2
6-3 Dividing PolynomialsExample 2B Continued
Draw a box around the remainder, 45.
3 –1 0 5 –1 –2
Step 3 Bring down the first coefficient. Then multiply and add for each column.
–63 45
Step 4 Write the quotient.
3x3 – 7x2 + 14x – 23 + 45x + 2
Write the remainder over the divisor.
46–2814–2314–7
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 2a
Divide using synthetic division.(6x2 – 5x – 6) ÷ (x + 3)
Step 1 Find a.
Write the coefficients of 6x2 – 5x – 6.
For (x + 3), a = –3.a = –3
–3 6 –5 –6
Step 2 Write the coefficients and a in the synthetic division format.
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 2a Continued
Draw a box around the remainder, 63.
6 –5 –6 –3
Step 3 Bring down the first coefficient. Then multiply and add for each column.
–18 6 63
Step 4 Write the quotient.
6x – 23 + 63x + 3
Write the remainder over the divisor.
–2369
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 2b
Divide using synthetic division.(x2 – 3x – 18) ÷ (x – 6)
Step 1 Find a.
Write the coefficients of x2 – 3x – 18.
For (x – 6), a = 6.a = 6
6 1 –3 –18
Step 2 Write the coefficients and a in the synthetic division format.
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 2b Continued
There is no remainder. 1 –3 –18 6
Step 3 Bring down the first coefficient. Then multiply and add for each column.
6 1 0
Step 4 Write the quotient.
x + 3
183
Holt McDougal Algebra 2
6-3 Dividing PolynomialsCheck It Out! Example 3
Write an expression for the length of a rectangle with width y – 9 and area y2 – 14y + 45.
Substitute.
Use synthetic division.
The area A is related to the width w and the length l by the equation A = l w.
y2 – 14y + 45 y – 9l(x) =
1 –14 45 9 9
1 0The length of the rectangle can be represented by l(x)= y – 5.
–45–5
Holt McDougal Algebra 2
6-3 Dividing Polynomials
3. Find an expression for the height of a parallelogram whose area is represented by 2x3 – x2 – 20x + 3 and whose base is represented by (x + 3).
Lesson Quiz
2. Divide by using synthetic division. (x3 – 3x + 5) ÷ (x + 2)
1. Divide by using long division. (8x3 + 6x2 + 7) ÷ (x + 2)
2x2 – 7x + 1
8x2 – 10x + 20 – 33 x + 2
x2 – 2x + 1 + 3 x + 2