Use long division to find the quotient below.
$\left(x^2+14 x+48\right)-(x+8)$
A. $x+6$
B. $x-6$
C. $x-40$
D. $x+40$
Answer (1)
Perform polynomial long division of x 2 + 14 x + 48 by x + 8 .
Divide x 2 by x to get the first term of the quotient, which is x .
Multiply ( x + 8 ) by x and subtract the result from x 2 + 14 x + 48 to get 6 x + 48 .
Divide 6 x by x to get the second term of the quotient, which is 6 .
The quotient is x + 6 .
Explanation
Understanding the Problem We are asked to divide the quadratic polynomial x 2 + 14 x + 48 by the linear binomial x + 8 using long division and determine the quotient.
First Step of Long Division To perform polynomial long division, we set up the division as follows:
x + ?
x + 8 | x^2 + 14x + 48
We ask: What do we multiply x by to get x 2 ? The answer is x . So, we write x above the 14 x term.
x x + 8 | x^2 + 14x + 48
Next, we multiply the divisor ( x + 8 ) by x to get x ( x + 8 ) = x 2 + 8 x . We subtract this from the dividend.
x x + 8 | x^2 + 14x + 48 - (x^2 + 8x) ------------- 6x + 48
Second Step of Long Division Now, we bring down the + 48 and ask: What do we multiply x by to get 6 x ? The answer is 6 . So, we write + 6 next to x in the quotient.
x + 6
x + 8 | x^2 + 14x + 48 - (x^2 + 8x) ------------- 6x + 48
Next, we multiply the divisor ( x + 8 ) by 6 to get 6 ( x + 8 ) = 6 x + 48 . We subtract this from 6 x + 48 .
x + 6 x + 8 | x^2 + 14x + 48 - (x^2 + 8x) ------------- 6x + 48 - (6x + 48) ------------- 0
Finding the Quotient Since the remainder is 0 , the division is exact, and the quotient is x + 6 .
Final Answer Therefore, x + 8 x 2 + 14 x + 48 = x + 6 .
Examples
Polynomial long division is used in various applications, such as simplifying complex algebraic expressions, solving equations, and designing control systems in engineering. For example, when designing a filter for signal processing, engineers use polynomial division to analyze the transfer function and ensure the filter meets specific performance criteria. This technique helps in breaking down complex systems into simpler, manageable components, facilitating analysis and design.
