The key here is the Remainder Theorem. This theorem states that for a polynomial p(x), the remainder of p(x) divided by (x-c) is exactly p(c). Thus, we can shift our problem of division (which can be complex and time-consuming), into a problem of evaluation (which is computation fast). For each of the expressions, we just need to replace 'x' with the value that makes the denominator zero and then compute the entire expression under this new value.
1. (12x-5x^4+231) / (x-2):
First, we set x=2, because 2 is the number that makes the denominator 'x-2' zero. We substitute, simplify and get 44, so the answer for the first expression is (e).
2. (5x^2-56) / (x-3):
For the second expression, we replace x with 3. Simplifying gives us 1, so the pair is answered with (d).
3. (5x^5+7x^3-4x^2-157) / (x-2):
Next, following the same principle, we replace x with 2. After the simplification, the answer is -11, so the match for item number three is (b).
4. (2x^7+5x^2+18) / (x-1):
We replace x with 1, which leads us to a final value of -53, so the answer to the fourth problem is (a).
5. (4x^3+x^2+7x-11) / (x+2):
Finally, for the last problem, we replace x with -2 (as '-2' makes 'x + 2' zero). Thus, the expression becomes -212 and its match is (c). And our matching is done.
Answers & Comments
Answer:
Answer【Answer】: 1. e. 2. d. 3.b. 4. a. 5. c.
【Explanation】:
The key here is the Remainder Theorem. This theorem states that for a polynomial p(x), the remainder of p(x) divided by (x-c) is exactly p(c). Thus, we can shift our problem of division (which can be complex and time-consuming), into a problem of evaluation (which is computation fast). For each of the expressions, we just need to replace 'x' with the value that makes the denominator zero and then compute the entire expression under this new value.
1. (12x-5x^4+231) / (x-2):
First, we set x=2, because 2 is the number that makes the denominator 'x-2' zero. We substitute, simplify and get 44, so the answer for the first expression is (e).
2. (5x^2-56) / (x-3):
For the second expression, we replace x with 3. Simplifying gives us 1, so the pair is answered with (d).
3. (5x^5+7x^3-4x^2-157) / (x-2):
Next, following the same principle, we replace x with 2. After the simplification, the answer is -11, so the match for item number three is (b).
4. (2x^7+5x^2+18) / (x-1):
We replace x with 1, which leads us to a final value of -53, so the answer to the fourth problem is (a).
5. (4x^3+x^2+7x-11) / (x+2):
Finally, for the last problem, we replace x with -2 (as '-2' makes 'x + 2' zero). Thus, the expression becomes -212 and its match is (c). And our matching is done.
In summary, we have the pairs 1-e, 2-d, 3-b, 4-a