Which polynomial is a factor of
$x^3 - 5x^2 + 11x - 10$?
A - HSC - SSCE Mathematics Extension 1 - Question 1 - 2017 - Paper 1
Question 1
Which polynomial is a factor of
$x^3 - 5x^2 + 11x - 10$?
A. $x - 2$
B. $x + 2$
C. $11x - 10$
D. $x^2 - 5x + 11$
Worked Solution & Example Answer:Which polynomial is a factor of
$x^3 - 5x^2 + 11x - 10$?
A - HSC - SSCE Mathematics Extension 1 - Question 1 - 2017 - Paper 1
Step 1
Determine if $x - 2$ is a factor
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Answer
To test if x−2 is a factor, substitute x=2 into the polynomial:
23−5(22)+11(2)−10=8−20+22−10=0
Since the result is 0, x−2 is indeed a factor.
Step 2
Determine if $x + 2$ is a factor
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Answer
Now substitute x=−2 into the polynomial:
(−2)3−5(−2)2+11(−2)−10=−8−20−22−10=−60
This result is not 0, so x+2 is not a factor.
Step 3
Determine if $11x - 10$ is a factor
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This calculation is complex, but checking quickly shows it does not return 0, meaning 11x−10 is not a factor.
Step 4
Determine if $x^2 - 5x + 11$ is a factor
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Answer
To check if x2−5x+11 can be a factor, we would perform polynomial long division. However, testing factors with synthetic division shows that this does not divide evenly as it yields a remainder.
