Lesson 6 – Dividing Polynomials by a Polynomial
 Dividing polynomials of various degrees by polynomials of various degrees
Dividing a polynomial by a polynomial
This process follows steps similar to those used in arithmetic.
Example 1
Divide 3 x 3  2 x 2  6 x  4 by x + 2
Solution
Always focus first on the leading terms.
Find how many times x goes into 3x3.
The result is 3x2.
Write the result in the x2 column on the solution line.
Now multiply the whole of x + 2 by 3x2.
The result is 3x3 + 6x2.
Subtract, giving a remainder of -4x2.
Bring down the -6x.
Now focus again on the leading terms.
Find how many times x goes into -4x2.
The result is -4x. Write this on the solution line.
Multiply the whole of x + 2 by -4x.
Subtract, giving a remainder of 2x.
Bring down the 4.
Find how many times x goes into 2x.
The result is 2. Write this on the solution line.
Multiply the whole of x + 2 by 2.
The result is 2x + 4.
Subtract.
The remainder is zero.
This means that x + 2 is a factor of the original cubic.
Example 2
Divide  4 x 3  6 x 2  4 x  7 by 2x – 3
Solution
−2x 2
+2
The quotient is  2x 2  2 with a remainder of –1.
Example 3
Divide x 4  25 x 2  62x  36 by x 2  3x  18
Solution
Support Question
1. Find each quotient and remainder. Assume the divisor is not equal to zero.


a. x 2  7 x  14  x  3




b. x 2  3 x  5  x  2


c. x 3  5 x 2  10 x  15  x  3
d. x 3  5x 2  x  10  x  2
e. (2x 2  1  5 x )  x  1
f.


g. x 3  10 x  15  7 x 2  x  8
i.
x 3  3 x 2  4 x  12
x 2
x

3

 13 x 2  39 x  20  x  9
 
h. x 3  5 x 2  2x  24  x 2  7 x  12

Key Question #6
1. Find each quotient and remainder. Assume the divisor is not equal to zero.



a. x 2  x  2  x  3

b. x 2  11x  6  x  5




c. 3x 3  11x 2  6x  10  x  4
d. 2x 3  x 2  27 x  36  x  3
e. (25x 2  1)  5 x  3
f.


3 x 3  2x 2  11x  12
x 1
2

 29 x  x 3  40   3  x 

g. 6  7 x  11x 2  2x 3  x  9
i.
2x
 
h. x 3  2x 2  4 x  15  x 2  2x  3



j. x 2  x 2 y  9 xy 2  9y 3  x  y 
2. When a certain polynomial is divided by x + 2, the quotient is x 2  4x  1 and the
remainder is 8. What is the polynomial?
3. When a certain polynomial is divided by x - 3, the quotient is x 2  2 x  5 and the
remainder is -3. What is the polynomial?
4. One factor of 4 x 3  15 x 2  31x  30 is x –2. Find the other factors.
5. When 10 x 3  mx 2  x  10 is divided by 5x – 3, the quotient is 2x 2  nx  2 and the
remainder is 4. Find the values of m and n.
6. Find the value of k such that when 2 x 3  9 x 2  kx  15 is divided by x+5, the remainder
is 0.