What is the remainder when $P(x) = -x^3 - 2x^2 - 3x + 8$ is divided by $x + 2$?
A - HSC - SSCE Mathematics Extension 1 - Question 3 - 2021 - Paper 1
Question 3
What is the remainder when $P(x) = -x^3 - 2x^2 - 3x + 8$ is divided by $x + 2$?
A. -14
B. -2
C. 2
D. 14
Worked Solution & Example Answer:What is the remainder when $P(x) = -x^3 - 2x^2 - 3x + 8$ is divided by $x + 2$?
A - HSC - SSCE Mathematics Extension 1 - Question 3 - 2021 - Paper 1
Step 1
Use the Remainder Theorem
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Answer
According to the Remainder Theorem, when a polynomial P(x) is divided by x−r, the remainder is P(r). Here, we're dividing by x+2, which can be rewritten as x−(−2), so r=−2.
Step 2
Evaluate $P(-2)$
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Answer
We need to evaluate the polynomial at x=−2:
P(−2)=−(−2)3−2(−2)2−3(−2)+8
Calculating each term:
The first term: −(−2)3=−(−8)=8.
The second term: −2(−2)2=−2(4)=−8.
The third term: −3(−2)=6.
The last term remains 8.
Combining these gives:
P(−2)=8−8+6+8=14
Step 3
Conclusion
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Answer
The remainder when P(x) is divided by x+2 is 14, which corresponds to option D.
