Roots of Polynomials (Edexcel A-Level Further Mathematics): Flashcards

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Roots of Polynomials
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Roots of polynomial

Values that satisfy f(x)=0f(x) = 0

Sum of roots S1S_1 for degree nn poly

S1=an1anS_1 = -\frac{a_{n-1}}{a_n}

Sum of products (2 roots) S2S_2 for degree nn

S2=an2anS_2 = \frac{a_{n-2}}{a_n}

Product of all roots PP for degree nn

P=(1)na0anP = (-1)^n \frac{a_0}{a_n}

α+β+γ\alpha + \beta + \gamma for ax3+bx2+cx+d=0ax^3+bx^2+cx+d=0

ba-\frac{b}{a}

αβ+βγ+γα\alpha\beta + \beta\gamma + \gamma\alpha for cubic

ca\frac{c}{a}

αβγ\alpha\beta\gamma for ax3+bx2+cx+d=0ax^3+bx^2+cx+d=0

da-\frac{d}{a}

α2+β2+γ2\alpha^2 + \beta^2 + \gamma^2 in terms of S1,S2S_1, S_2

S122S2S_1^2 - 2S_2

1α+1β+1γ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} formula

S2P\frac{S_2}{P}

(c+α)(c+β)(c+γ)(c+\alpha)(c+\beta)(c+\gamma) expanded

c3+S1c2+S2c+Pc^3 + S_1c^2 + S_2c + P

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