← Complex numbers: past papers
AS 91577 · 2024 paper · a factor, a power in polar form, a surd equation, a locus gradient, and a line a quotient never meets
Question 1(a) Achieved
Practises Polynomial equations with complex roots
Five separate parts: a factor of a cubic, a power of a number in polar form, an equation with two square roots, the gradient of a straight-line locus, and a proof that a quotient never lies on y = x.
Given that x + 3 is a factor of x3 + px2 + 5x − 12, find the value of the constant p.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 If x + 3 is a factor, what is the remainder when you divide by it?
Zero. By the factor theorem, f(−3) = 0: the value that makes x + 3 zero makes the whole cubic zero. -
2 Put x = −3 in and solve for p.
(−3)3 + p(−3)2 + 5(−3) − 12 = 0
−27 + 9p − 15 − 12 = 0
9p = 54, so p = 6.
What the marker wanted
- Achieved The value of p, from f(−3) = 0.
In our words, from NZQA’s 2024 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.