Your answer should be in the form $p(x)+\frac{k}{x+3}$ where $p$ is a polynomial and $k$ is an integer.
$\frac{x^2-7}{x+3}=$
Answer (1)
Perform polynomial long division of x 2 − 7 by x + 3 .
Find the quotient and remainder.
Express the result in the form p ( x ) + x + 3 k .
The final answer is x − 3 + x + 3 2 .
Explanation
Understanding the Problem We are given the expression x + 3 x 2 − 7 and we want to express it in the form p ( x ) + x + 3 k , where p ( x ) is a polynomial and k is an integer. This means we need to perform polynomial long division.
Polynomial Long Division We perform polynomial long division of x 2 − 7 by x + 3 .
Performing the Division Dividing x 2 − 7 by x + 3 , we get a quotient of x − 3 and a remainder of 2 . This can be verified as follows: ( x + 3 ) ( x − 3 ) + 2 = x 2 − 9 + 2 = x 2 − 7
Expressing the Result Therefore, we can write x + 3 x 2 − 7 = x − 3 + x + 3 2 .
Final Answer Thus, p ( x ) = x − 3 and k = 2 . The expression in the required form is x − 3 + x + 3 2 .
Examples
Polynomial division is a fundamental concept in algebra and is used to simplify complex rational expressions. For example, in physics, when analyzing the motion of objects under variable forces, you might encounter expressions involving ratios of polynomials. Simplifying these expressions using polynomial division can make the analysis easier and reveal underlying relationships between variables. Also, polynomial division is used in calculus when integrating rational functions.
