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NSWMaths Extension 1Quick questions

Polynomials (ME-F2)

Quick questions on Consequences of the factor theorem: distinct zeroes give distinct factors, a degree-n polynomial has at most n zeroes, agreement at n + 1 points forces identity, a graph is determined by n + 1 points, intersection counts are bounded by degree via P(x) - Q(x), and re-expressing a polynomial in powers of (x - a)

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are distinct zeroes give distinct factors?
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Suppose you have found several different zeroes of a polynomial P(x)P(x), say α1,α2,…,αs\alpha_1, \alpha_2, \dots, \alpha_s, all distinct. The factor theorem makes each one a factor: x−α1x - \alpha_1 divides P(x)P(x), so P(x)=(x−α1)P1(x)P(x) = (x - \alpha_1)P_1(x). Now α2\alpha_2 is also a zero, so P(α2)=0P(\alpha_2) = 0; but (α2−α1)≠0(\alpha_2 - \alpha_1) \ne 0 because the zeroes are distinct, so the other factor P1(α2)P_1(\alpha_2) must be zero. That makes x−α2x - \alpha_2 a factor of P1(x)P_1(x), and continuing this way each distinct zero peels off its own linear factor.
What is finding all the zeroes when they are distinct?
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The first bound turns into a fast factoring method whenever a polynomial has only simple (distinct) zeroes. Instead of finding one factor, dividing, and repeating, you simply keep testing divisors of the constant term until you have collected as many distinct zeroes as the degree.
What is counting intersections with the difference F(x) = P(x) - Q(x)?
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The single most useful tool on this page is the difference function F(x)=P(x)−Q(x)F(x) = P(x) - Q(x). Two curves y=P(x)y = P(x) and y=Q(x)y = Q(x) meet exactly where P(x)=Q(x)P(x) = Q(x), that is, exactly where F(x)=0F(x) = 0. So the intersections of the two curves are precisely the zeroes of the one polynomial FF, and the whole intersection question collapses to factoring FF.
What is re-expressing a polynomial in powers of (x - a)?
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The identity condition also justifies a useful change of viewpoint: any polynomial can be rewritten in powers of (x−a)(x - a) rather than powers of xx. For a cubic and aa a chosen number, you write

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