Use long division to find the quotient below.
$\left(x^5+18 x^2-27 x\right) \div(x+3)$
A. $x^4+x^3-6 x^2-9 x$
B. $x^4-3 x^3+9 x^2-9 x$
C. $x^4+3 x^3-9 x^2-9 x$
D. $x^4-x^3+6 x^2-9 x$
Answer (1)
Set up the polynomial long division with x 5 + 0 x 4 + 0 x 3 + 18 x 2 − 27 x + 0 as the dividend and x + 3 as the divisor.
Perform the long division step by step, dividing, multiplying, and subtracting until the remainder is obtained.
After performing the long division, identify the quotient.
The quotient of the division is x 4 − 3 x 3 + 9 x 2 − 9 x .
Explanation
Understanding the Problem We are asked to perform polynomial long division to find the quotient of ( x 5 + 18 x 2 − 27 x ) ÷ ( x + 3 ) . This means we want to divide the polynomial x 5 + 18 x 2 − 27 x by the polynomial x + 3 .
Setting up Long Division We set up the long division as follows:
x + 3 | x^5 + 0x^4 + 0x^3 + 18x^2 - 27x + 0
Note that we include the terms 0 x 4 , 0 x 3 , and 0 to keep track of the powers of x .
First Iteration
Divide x 5 by x to get x 4 . Multiply x 4 by ( x + 3 ) to get x 5 + 3 x 4 . Subtract this from the dividend to get − 3 x 4 + 0 x 3 + 18 x 2 − 27 x + 0 .
x^4
x + 3 | x^5 + 0x^4 + 0x^3 + 18x^2 - 27x + 0
-(x^5 + 3x^4)
------------------
-3x^4 + 0x^3
Second Iteration
Divide − 3 x 4 by x to get − 3 x 3 . Multiply − 3 x 3 by ( x + 3 ) to get − 3 x 4 − 9 x 3 . Subtract this from the remaining dividend to get 9 x 3 + 18 x 2 − 27 x + 0 .
x^4 - 3x^3
x + 3 | x^5 + 0x^4 + 0x^3 + 18x^2 - 27x + 0
-(x^5 + 3x^4)
------------------
-3x^4 + 0x^3
-(-3x^4 - 9x^3)
------------------
9x^3 + 18x^2
Third Iteration
Divide 9 x 3 by x to get 9 x 2 . Multiply 9 x 2 by ( x + 3 ) to get 9 x 3 + 27 x 2 . Subtract this from the remaining dividend to get − 9 x 2 − 27 x + 0 .
x^4 - 3x^3 + 9x^2
x + 3 | x^5 + 0x^4 + 0x^3 + 18x^2 - 27x + 0
-(x^5 + 3x^4)
------------------
-3x^4 + 0x^3
-(-3x^4 - 9x^3)
------------------
9x^3 + 18x^2
-(9x^3 + 27x^2)
------------------
-9x^2 - 27x
Fourth Iteration
Divide − 9 x 2 by x to get − 9 x . Multiply − 9 x by ( x + 3 ) to get − 9 x 2 − 27 x . Subtract this from the remaining dividend to get 0 .
x^4 - 3x^3 + 9x^2 - 9x
x + 3 | x^5 + 0x^4 + 0x^3 + 18x^2 - 27x + 0
-(x^5 + 3x^4)
------------------
-3x^4 + 0x^3
-(-3x^4 - 9x^3)
------------------
9x^3 + 18x^2
-(9x^3 + 27x^2)
------------------
-9x^2 - 27x
-(-9x^2 - 27x)
------------------
0
Final Result The quotient is x 4 − 3 x 3 + 9 x 2 − 9 x and the remainder is 0 .
Conclusion Therefore, ( x 5 + 18 x 2 − 27 x ) ÷ ( x + 3 ) = x 4 − 3 x 3 + 9 x 2 − 9 x .
Examples
Polynomial long division isn't just an abstract math concept; it has practical applications in various fields. For instance, in engineering, when designing systems, you might model different components using polynomials. If you need to simplify the model or analyze a specific part, polynomial division can help break down complex expressions into simpler, manageable forms. This allows engineers to understand the behavior of the system more effectively and make informed decisions about its design and optimization. By dividing one polynomial (representing the entire system) by another (representing a component), you can isolate and study the remaining parts.
