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NSWMaths Extension 1Quick questions

Functions (ME-F1, ME-F2)

Quick questions on Roots and coefficients of polynomials: Vieta's formulas for cubics and quartics

11short 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 is vieta's formulas?
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For a polynomial of degree nn with leading coefficient ana_n and roots α1,α2,…,αn\alpha_1, \alpha_2, \dots, \alpha_n (counted with multiplicity),
What is specialised formulas?
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Quadratic ax2+bx+ca x^2 + b x + c with roots α,β\alpha, \beta:
What is constructing a polynomial from its roots?
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If the roots are α1,…,αn\alpha_1, \dots, \alpha_n, the monic polynomial with these roots is
What is useful identities involving symmetric functions?
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For roots α,β,γ\alpha, \beta, \gamma of a cubic,
What is cubic with one given root?
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The cubic x3+px+q=0x^3 + p x + q = 0 has roots α,β,γ\alpha, \beta, \gamma. If α=1\alpha = 1, β+γ=2\beta + \gamma = 2 and βγ=−3\beta \gamma = -3, find pp and qq.
What is building a quartic?
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Find a monic quartic with roots 1,2,3,41, 2, 3, 4.
What is sum of squares of roots?
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The cubic x3−3x2+4x−1x^3 - 3 x^2 + 4 x - 1 has roots α,β,γ\alpha, \beta, \gamma. Find α2+β2+γ2\alpha^2 + \beta^2 + \gamma^2.
What is sign errors on the k ^k alternation?
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Sum is negative of the next coefficient, sum-in-pairs is plus the one after, sum-in-triples is negative again. Track the signs carefully.
What is confusing the third symmetric function with a sum?
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For a quartic, ∑triples\sum_{\text{triples}} adds four terms each a product of three roots, not the sum of every triple of three roots multiplied together.
What is using Vieta where roots are not allowed to be repeated?
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Vieta's formulas count roots with multiplicity. A double root contributes twice to the sum.
What is equating coefficient signs without checking?
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When comparing P(x)=x3+bx2+cx+dP(x) = x^3 + b x^2 + c x + d with sum-product information, always restate which Vieta gives which coefficient. :::

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