Divide the polynomials.
Your answer should be in the form [tex]$p(x)+\frac{k}{x+5}$[/tex] where [tex]$p$[/tex] is a polynomial and [tex]$k$[/tex] is an integer.
[tex]$\frac{x^2-28}{x+5}=$\square[/tex]
Answer (1)
Perform polynomial long division to divide x 2 − 28 by x + 5 .
Identify the quotient and remainder from the long division.
Express the result in the form p ( x ) + x + 5 k .
The final answer is x − 5 + x + 5 − 3 .
Explanation
Understanding the Problem We are asked to divide the polynomial x 2 − 28 by x + 5 and express the result in the form p ( x ) + x + 5 k , where p ( x ) is a polynomial and k is an integer. This means we need to perform polynomial division and identify the quotient p ( x ) and the remainder k .
Performing Polynomial Long Division To divide x 2 − 28 by x + 5 , we can use polynomial long division. Alternatively, we can use synthetic division or manipulate the expression algebraically.
Let's perform polynomial long division:
x - 5
x+5 | x^2 + 0x - 28 - (x^2 + 5x) ------------ -5x - 28 - (-5x - 25) ------------ -3
So, the quotient is x − 5 and the remainder is − 3 .
Expressing the Result in the Required Form Therefore, we can write the given expression as: x + 5 x 2 − 28 = x − 5 + x + 5 − 3 = x − 5 − x + 5 3
Thus, p ( x ) = x − 5 and k = − 3 .
Verifying the Result To verify our result, we can multiply ( x − 5 ) by ( x + 5 ) and add − 3 :
( x − 5 ) ( x + 5 ) − 3 = x 2 − 25 − 3 = x 2 − 28
This confirms that our division is correct.
Final Answer The result of dividing x 2 − 28 by x + 5 is x − 5 − x + 5 3 . Therefore, p ( x ) = x − 5 and k = − 3 .
The answer is x − 5 + x + 5 − 3 .
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, engineers use polynomial division to analyze the stability of control systems. Imagine designing a bridge; engineers use polynomials to model the forces and stresses acting on the structure. By dividing these polynomials, they can determine critical points and ensure the bridge's stability under different loads. This ensures safety and prevents potential collapses.
