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
Identify the expression to factor
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Answer
We start with the polynomial expression:
x3−5x2+11x−10.
Step 2
Test possible factors
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Answer
We will substitute the values from the options into the polynomial to see which one results in zero.
Testing x=2 (option A):
23−5(22)+11(2)−10=8−20+22−10=0
Thus, x−2 is a factor.
Testing x=−2 (option B):
(−2)3−5(−2)2+11(−2)−10=−8−20−22−10=−60
Not a factor.
Testing x = rac{10}{11} (option C):
This does not yield a simple integer, making it less likely to be a factor.
Testing the quadratic x2−5x+11 (option D) would need further polynomial long division to be verified.
Step 3
Conclusion
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Answer
Based on the tests, the polynomial that is a factor of the original expression is:
