ENGINEERING ALGEBRA
Lesson 6:
Division of Polynomials
Dividing a polynomial by another polynomial with the same or lower degree is called
“division of polynomials”. It reduces the complex division into smaller one using a long
division of polynomials or short/synthetic division.
Simple Division
-
Dividing a polynomial by a monomial.
6r 2 s 2  3rs 2  9r 2 s
1.
3rs
6r 2 s2 3rs2 9r 2 s



3rs
3rs
3rs
 2rs  s  3r
3a 2b  6a 3b 2  18ab
2.
3ab

3a2 b 6a 3b 2 18ab


3ab
3ab
3ab
2
 a  2a b  6
3.
12 x 2 y  3 x
3x
12x 2 y 3x


3x
3x
 4xy1
Long Division of Polynomials
-
Divide a polynomial by a polynomial
Consider a polynomial 𝑥 3 − 4𝑥 2 − 8 divided by 𝑥 − 3. The divisor is 𝑥 − 3 and;
The dividend is 𝑥 3 − 4𝑥 2 − 8, which should be arranged in descending order, such as
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𝑥 3 − 4𝑥 2 + 0𝑥 − 8
To find for the quotient and the remainder; the procedures are as follows;
Step 1: Divide the first term of the dividend by the first term of the divisor, and place the
result on top of the bar.
𝑥2
𝑥 − 3√𝑥 3 − 4𝑥 2 + 0𝑥 − 8
Step 2: Multiply the result by each term of the divisor, and place the product below the
dividend.
𝑥2
𝑥 − 3√𝑥 3 − 4𝑥 2 + 0𝑥 − 8
𝑥 3 − 3𝑥 2
Step 3: Subtract the obtained product from the upper terms, be careful with the sign of the
subtrahend it will be reversed, then bring down the next term of the dividend.
𝑥2
𝑥 − 3√𝑥 3 − 4𝑥 2 + 0𝑥 − 8
−𝑥 3 + 3𝑥 2
−𝑥 2 + 0𝑥
Step 4: Repeat steps 1 to 3, until there’s no term left from the dividend to bring down.
𝑥2 − 𝑥 − 3
𝑥 − 3√𝑥 3 − 4𝑥 2 + 0𝑥 − 8
−𝑥 3 + 3𝑥 2
−𝑥 2 + 0𝑥
−𝑥 2 + 3𝑥
−3𝑥 − 8
Therefore:
−3𝑥 + 9
𝑥 2 − 𝑥 − 3 Is the quotient and
-17 is the remainder.
Ans.
−17
Check:
𝑥 3 − 4𝑥 2 − 8 = (𝑥 − 3)(𝑥 2 − 𝑥 − 3) − 17
= 𝑥 3 − 𝑥 2 − 3𝑥 − 3𝑥 2 + 3𝑥 + 9 − 17
𝑥 3 − 4𝑥 2 − 8 = 𝑥 3 − 4𝑥 2 − 8
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Short/Synthetic Division of Polynomials
-
Divide a polynomial by a polynomial.
To use synthetic division:


There must be a coefficient for every possible power of the variable.
The divisor must have a leading coefficient of 1.
1. Consider a polynomial 𝑥 3 − 4𝑥 2 − 8 divided by 𝑥 − 3 to find for the quotient and the
remainder; the procedures are as follows;
Step 1: Arranged numerator in descending order.
𝑥 3 − 4𝑥 2 + 0𝑥 − 8 / 𝑥 − 3
Step 2: Place the coefficients inside, according to their order.
1
-4
0
-8
Step 3: Place the numerical of the denominator outside with opposite sign.
+3
1
-4
0
Note: synthetic division can only
be done when there is a first
degree x in the denominator
-8
Step 4: Bring down 1st term below the bar. Then multiply 3 by 1 then placed the product
below -4 and add, placed the sum next to 1.
+3
1
-4
0
-8
3
1
-1
Step 5: Multiply 3 by -1 then placed the product below 0 and add, placed the sum next to 1. Repeat this process up to the last term.
+3
1
1
-4
0
-8
3
-3
-9
-1 -3
-17
Remainder
The result should be one degree lesser than the given polynomial, and the numbers
below the bar such as 1, -1 & -3 are the resulting coefficients. Therefore:
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𝑥 3 −4𝑥 2 −8
𝑥−3
= (𝑥 2 − 𝑥 − 3) −
17
𝑥−3
Ans.
To check: Cross multiply the denominator of the left side of the equation to the
right side, then simplify.
𝑥 3 − 4𝑥 2 − 8 = (𝑥 − 3)(𝑥 2 − 𝑥 − 3) − 17
= 𝑥 3 − 𝑥 2 − 3𝑥 − 3𝑥 2 + 3𝑥 + 9 − 17
𝑥 3 − 4𝑥 2 − 8 = 𝑥 3 − 4𝑥 2 − 8
2.
x
3
 x  2x  7  2x  1
2
 x3 x 2 2 x 7   2 x 1 



 
 
2
2 2  2 2
 2
7 
1
1 3 1 2
 x  x  x   x  
2
2 
2
2
1
1
2
2

1
2
1

1
7
2

1
7
4
16
8
_________________________
1
2

1
7
49
4
8
16
Suggested Video/s:
1. https://www.youtube.com/watch?v=_FSXJmESFmQ
2. https://www.youtube.com/watch?v=FxHWoUOq2iQ
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