1. EXAMPLE 1
Use polynomial long division
Divide f (x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5.
SOLUTION
Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

2. ) EXAMPLE 1 Use polynomial long division 3x2 + 4x – 3 x2 – 3x + 5
quotient
x2 – 3x + 5
3x4 – 5x3 + 0x2 + 4x – 6 )
Multiply divisor by 3x4/x2 = 3x2
3x4 – 9x3 + 15x2
Subtract. Bring down next term.
4x3 – 15x x
4x3 – 12x2 + 20x
Multiply divisor by 4x3/x2 = 4x
Subtract. Bring down next term.
– 3x2 – 16x – 6
–3x2 + 9x – 15
Multiply divisor by – 3x2/x2 = – 3
– 25x + 9
remainder

3. EXAMPLE 1
Use polynomial long division
3x4 – 5x3 + 4x – 6
x2 – 3x + 5
= 3x2 + 4x – 3 +
– 25x + 9
ANSWER
CHECK
You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend.
(3x2 + 4x – 3)(x2 – 3x + 5) + (– 25x + 9)
= 3x2(x2 – 3x + 5) + 4x(x2 – 3x + 5) – 3(x2 – 3x + 5) – 25x + 9
= 3x4 – 9x3 + 15x2 + 4x3 – 12x2 + 20x – 3x2 + 9x – 15 – 25x + 9
= 3x4 – 5x3 + 4x – 6

4. ) EXAMPLE 2 Use polynomial long division with a linear divisor
Divide f (x) = x3 + 5x2 – 7x + 2 by x – 2.
x2 + 7x
quotient
x – 2
x3 + 5x2 – 7x )
x3 – 2x2
Multiply divisor by x3/x = x2.
7x2 – 7x
Subtract.
7x2 – 14x
Multiply divisor by 7x2/x = 7x.
7x + 2
Subtract.
7x – 14
Multiply divisor by 7x/x = 7. 16
remainder
ANSWER
x3 + 5x2 – 7x +2
x – 2
= x2 + 7x + 7 + 16

5. GUIDED PRACTICE
for Examples 1 and 2
Divide using polynomial long division.
(2x4 + x3 + x – 1) (x2 + 2x – 1)
SOLUTION
Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

6. ) GUIDED PRACTICE for Examples 1 and 2 2x2 – 3x + 8 x2 + 2x – 1
quotient
x2 + 2x – 1
2x4 + x3 + 0x2 + x – 1 )
Multiply divisor by 2x4/x2 = –2x2.
2x4 – 4x3 – 2x2
Subtract. Bring down next term.
3x3 – 2x2 + x
– 3x3 – 6x2 + 3x
Multiply divisor by –3x3/x2 = –3.
Subtract. Bring down next term.
8x2 – 2x – 1
8x2 –16x – 8
Multiply divisor by 4x2/x2 = 8.
– 18x + 7
remainder

7. GUIDED PRACTICE
for Examples 1 and 2
2x4 + 5x3 + x – 1
x2 + 2x – 1
= (2x2 – 3x + 8)+
– 18x + 7
ANSWER

8. GUIDED PRACTICE
for Examples 1 and 2
2. (x3 – x2 + 4x – 10)  (x + 2)
SOLUTION
Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

9. ) GUIDED PRACTICE for Examples 1 and 2 x2 – 3x + 10 x + 2
quotient
x + 2
x3 – x2 + 4x – 10 )
Multiply divisor by x3/x = x2.
x3 + 2x2
Subtract. Bring down next term.
–3x2 + 4x
– 3x2 – 6x
Multiply divisor by –3x2/x = –3x.
Subtract. Bring down next term.
10x – 1
10x + 20
Multiply divisor by 10x/x = 10.
– 30
remainder

10. GUIDED PRACTICE
for Examples 1 and 2
x3 – x2 +4x – 10
x + 2
= (x2 – 3x +10)+
– 30
ANSWER