Rewrite the expression $\frac{x^2+25 x+3}{x+3}$ in the form $q(x)+\frac{r(x)}{b(x)}$ where $q(x)=$ quotient, $r(x)=$ remainder, and $b(x)=$ divisor, using the long division method.
A. $x+28+\frac{87}{x-3}$
B. $x+3+\frac{87}{x-28}$
C. $x-3+\frac{87}{x+3}$
D. $x+30+\frac{87}{x-3}$
Answer (2)
Using polynomial long division, we find that x + 3 x 2 + 25 x + 3 rewrites to x + 22 − x + 3 63 . However, none of the provided multiple-choice options match this result, suggesting a possible error in the options. The correct expression includes a quotient of x + 22 and a remainder of − 63 .
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Perform polynomial long division of x 2 + 25 x + 3 by x + 3 .
Divide x 2 by x to get x , multiply ( x + 3 ) by x to get x 2 + 3 x , and subtract from x 2 + 25 x + 3 to get 22 x + 3 .
Divide 22 x by x to get 22 , multiply ( x + 3 ) by 22 to get 22 x + 66 , and subtract from 22 x + 3 to get − 63 .
The result of the long division is x + 22 − x + 3 63 , which does not match any of the provided options, suggesting a possible typo in the options.
Explanation
Problem Analysis We are given the expression x + 3 x 2 + 25 x + 3 and asked to rewrite it in the form q ( x ) + b ( x ) r ( x ) , where q ( x ) is the quotient, r ( x ) is the remainder, and b ( x ) is the divisor. In this case, the divisor is b ( x ) = x + 3 . We will use polynomial long division to find the quotient and remainder.
Polynomial Long Division We perform polynomial long division of x 2 + 25 x + 3 by x + 3 . First, we divide x 2 by x to get x . Then, we multiply ( x + 3 ) by x to get x 2 + 3 x . Subtracting this from x 2 + 25 x + 3 gives ( x 2 + 25 x + 3 ) − ( x 2 + 3 x ) = 22 x + 3 . Next, we divide 22 x by x to get 22 . Then, we multiply ( x + 3 ) by 22 to get 22 x + 66 . Subtracting this from 22 x + 3 gives ( 22 x + 3 ) − ( 22 x + 66 ) = − 63 . Thus, the quotient is x + 22 and the remainder is − 63 .
Finding Quotient and Remainder Therefore, we can write the expression as x + 22 + x + 3 − 63 = x + 22 − x + 3 63 . However, this result does not match any of the given options. Let's re-examine the long division.
Verifying Long Division Let's perform the long division again:
x + 22
x+3 | x^2 + 25x + 3 -(x^2 + 3x) ---------- 22x + 3 -(22x + 66) ---------- -63
So, we have x 2 + 25 x + 3 = ( x + 3 ) ( x + 22 ) − 63 . Dividing by x + 3 , we get x + 3 x 2 + 25 x + 3 = x + 22 − x + 3 63 . This still doesn't match any of the options.
Checking the Options Let's check the options by multiplying them out. We are looking for an expression of the form q ( x ) + x + 3 r ( x ) such that when we combine the terms, we get x + 3 x 2 + 25 x + 3 .
Option 1: x + 28 + x − 3 87 is not in the correct form. Option 2: x + 3 + x − 28 87 is not in the correct form. Option 3: x − 3 + x + 3 87 = x + 3 ( x − 3 ) ( x + 3 ) + 87 = x + 3 x 2 − 9 + 87 = x + 3 x 2 + 78 . This is not equal to x + 3 x 2 + 25 x + 3 .
Option 4: x + 22 − x + 3 63 is what we found with long division. However, none of the options match this. Let's try x + 30 + x − 3 87 which is also not in the correct form.
Final Verification Let's reconsider the long division one more time. We have:
x + 22
x+3 | x^2 + 25x + 3 -(x^2 + 3x) ---------- 22x + 3 -(22x + 66) ---------- -63
So, x + 3 x 2 + 25 x + 3 = x + 22 − x + 3 63 . It seems there might be a typo in the options. However, let's try to manipulate our result to see if we can match any of the options. None of the options are in the form x + 22 + x + 3 co n s t an t .
Conclusion Let's try to find an error in the problem statement or the options. Assuming the problem statement is correct, let's look at the long division again. The long division is correct, resulting in x + 22 − x + 3 63 . However, none of the options match this result. It is possible that there is a typo in the provided options. If the question was x + 3 x 2 + 25 x + 3 , then the correct answer should be x + 22 − x + 3 63 . Since none of the options match, we will assume there is a typo in the options.
Examples
Polynomial long division is used in various applications, such as simplifying complex rational expressions in calculus, designing control systems in engineering, and optimizing resource allocation in economics. For example, if you have a function representing the cost of producing a certain item and another function representing the number of items produced, polynomial long division can help you determine the average cost per item as the number of items increases. This technique is also crucial in cryptography for tasks like error detection and correction in data transmission.
