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Holt Algebra 2
6-3 Dividing Polynomials
Polynomials can be divided using long division just
like you learned with numbers.
Divide) 214 ÷ 6
6 214 35
-18
34 -30
4 Remainder
214 ÷ 6 = 354
6
Holt Algebra 2
6-3 Dividing Polynomials
Holt 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, including terms 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 Algebra 2
6-3 Dividing Polynomials
Notice that y times 2y2 is 2y3.
Write 2y2 above 2y3.
Step 3 Divide.
2y2
–(2y3 – 6y2) Multiply y – 3 by 2y2. Then
subtract. Bring down the next
term. Divide 5y2 by y. 5y2 + 0y
+ 5y
–(5y2 – 15y) Multiply y – 3 by 5y. Then
subtract. Bring down the next
term. Divide 15y by y. 15y + 25
–(15y – 45)
70 Find the remainder.
+ 15
Multiply y – 3 by 15. Then
subtract.
Example 1 Continued
y – 3 2y3 – y2 + 0y + 25
Holt Algebra 2
6-3 Dividing Polynomials
Step 4 Write the final answer.
Example 1 Continued
–y2 + 2y3 + 25
y – 3 = 2y2 + 5y + 15 + 70
y – 3
Holt Algebra 2
6-3 Dividing Polynomials
Check It Out! Example 1a
Divide using long division.
(15x2 + 8x – 12) ÷ (3x + 1)
15x2 + 8x – 12
Step 1 Write the dividend in standard form, including terms with a coefficient of 0.
Step 2 Write division in the same way you would when dividing numbers.
3x + 1 15x2 + 8x – 12
Holt Algebra 2
6-3 Dividing Polynomials
Check It Out! Example 1a Continued
Notice that 3x times 5x is 15x2.
Write 5x above 15x2.
Step 3 Divide.
5x
–(15x2 + 5x) Multiply 3x + 1 by 5x. Then
subtract. Bring down the next
term. Divide 3x by 3x. 3x – 12
+ 1
–(3x + 1)
–13 Find the remainder.
Multiply 3x + 1 by 1. Then
subtract.
3x + 1 15x2 + 8x – 12
Holt Algebra 2
6-3 Dividing Polynomials
Check It Out! Example 1a Continued
Step 4 Write the final answer.
15x2 + 8x – 12
3x + 1 = 5x + 1 – 13
3x + 1
Holt Algebra 2
6-3 Dividing Polynomials
Check It Out! Example 1b
Divide using long division.
(x2 + 5x – 28) ÷ (x – 3)
x2 + 5x – 28
Step 1 Write the dividend in standard form, including terms with a coefficient of 0.
Step 2 Write division in the same way you would when dividing numbers.
x – 3 x2 + 5x – 28
Holt Algebra 2
6-3 Dividing Polynomials
Notice that x times x is x2.
Write x above x2.
Step 3 Divide.
x
–(x2 – 3x) Multiply x – 3 by x. Then
subtract. Bring down the next
term. Divide 8x by x. 8x – 28
+ 8
–(8x – 24)
–4 Find the remainder.
Multiply x – 3 by 8. Then
subtract.
Check It Out! Example 1b Continued
x – 3 x2 + 5x – 28
Holt Algebra 2
6-3 Dividing Polynomials
Check It Out! Example 1b Continued
Step 4 Write the final answer.
x2 + 5x – 28
x – 3 = x + 8 –
4
x – 3
Holt Algebra 2
6-3 Dividing Polynomials
Synthetic division is a shortcut method for dividing a polynomial by a linear binomial (x – a). The divisor must be a binomial. The x must have a coefficient of 1. Use the opposite sign of “a.”
Holt Algebra 2
6-3 Dividing Polynomials
Holt 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.
1 3
For (x – ), a = . 1 3
1 3
1 3
a =
1 3
3 9 –2
Holt Algebra 2
6-3 Dividing Polynomials
Example 2A Continued
Step 2 Bring down the first coefficient. Then multiply and add for each column.
Draw a box around the remainder, 1 . 1 3
1 3
3 9 –2
1
3
Step 3 Write the quotient.
3x + 10 + 1
1 3 1 3
x –
10 1 3
1
1 3
3
Holt Algebra 2
6-3 Dividing Polynomials
Example 2A Continued
3x + 10 + 1
1 3 1 3
x – Check Multiply (x – )
1 3
= 3x2 + 9x – 2
(x – ) 1 3
(x – ) 1 3
(x – ) 1 3
3x + 10 +
1 1 3 1 3
x –
Holt 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 Algebra 2
6-3 Dividing Polynomials
Example 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.
–6
3 45
Step 4 Write the quotient.
3x3 – 7x2 + 14x – 23 + 45
x + 2 Write the remainder over
the divisor.
46 –28 14
–23 14 –7
Holt Algebra 2
6-3 Dividing Polynomials
Check 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 Algebra 2
6-3 Dividing Polynomials
Check 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 + 63
x + 3 Write the remainder over
the divisor.
–23
69
Holt Algebra 2
6-3 Dividing Polynomials
Check 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 Algebra 2
6-3 Dividing Polynomials
Check 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
18
3
Holt Algebra 2
6-3 Dividing Polynomials
Homework pg. 426
#’s 19-29 odd, 37-43 odd