Core Pure 1 Chapter 4: Roots of Polynomials

14 flashcards · Mathematics · by TheoTyson

1. If α\alphaα and β\betaβ are roots of the equation ax2 + bx + c=0ax^2 \, + \, bx \, + \, c = 0ax2+bx+c=0 , how do you find the equation?

   α + β=−ba\alpha \, + \, \beta = - \frac{b}{a}α+β=−ab​ αβ=ca\alpha \beta = \frac{c}{a}αβ=ac​

2. If α, β\alpha, \, \betaα,β and γ\gammaγ are roots of the equation ax3 + bx2, + cx + d=0ax^3 \, + \, bx^2, \, + \, cx \, + \, d = 0ax3+bx2,+cx+d=0 , how do you find the equation?

   α + β + γ=−ba\alpha \, + \, \beta \, + \, \gamma = - \frac{b}{a}α+β+γ=−ab​ αβ + βγ + γα=ca\alpha \beta \, + \, \beta \gamma \, + \, \gamma \alpha = \frac{c}{a}αβ+βγ+γα=ac​ αβγ=−da\alpha \beta \gamma = - \frac{d}{a}αβγ=−ad​

3. If α, β, γ\alpha , \, \beta, \, \gammaα,β,γ and δ\deltaδ are roots of the equation ax4 + bx3 + cx2 + dx + e=0ax^4 \, + \, bx^3 \, + \, cx^2 \, + \, dx \, + \, e = 0ax4+bx3+cx2+dx+e=0 , how do you find the equation?

   α + β + γ + δ=−ba\alpha \, + \, \beta \, + \, \gamma \, + \, \delta = - \frac{b}{a}α+β+γ+δ=−ab​ αβ + αγ + αδ + βγ + βδ + γδ=ca\alpha \beta \, + \, \alpha \gamma \, + \, \alpha \delta \, + \, \beta \gamma \, + \, \beta \delta \, + \, \gamma \delta = \frac{c}{a}αβ+αγ+αδ+βγ+βδ+γδ=ac​ αβγ + αβδ + αγδ&T…

4. What is 1α+1β\frac{1}{\alpha} + \frac{1}{\beta}α1​+β1​ ?

   1α+1β=α + βαβ\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha \, + \, \beta}{\alpha \beta}α1​+β1​=αβα+β​

5. What is 1α+1β+1γ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}α1​+β1​+γ1​ ?

   1α+1β+1γ=αβ + βγ + γααβγ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\alpha \beta \, + \, \beta \gamma \, + \, \gamma \alpha}{\alpha \beta \gamma}α1​+β1​+γ1​=αβγαβ+βγ+γα​

6. What is 1α+1β+1γ+1δ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}α1​+β1​+γ1​+δ1​ ?

   1α+1β+1γ+1δ=αβγ + βγδ + γδα + δαβαβγδ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}= \frac{\alpha \beta \gamma \, + \, \beta \gamma \delta \, + \, \gamma \delta \alpha\, + \, \delta \alpha \beta}{\alpha \beta \gamma \delta}α1​+β1​+γ1​+δ1​=αβγδαβγ+βγδ+γδα+δαβ​

7. What is αn×βn\alpha^n \times \beta^nαn×βn ?

   αn×βn=(αβ)n\alpha^n \times \beta^n = (\alpha \beta)^nαn×βn=(αβ)n

8. What is αn×βn×γn\alpha^n \times \beta^n \times \gamma^nαn×βn×γn ?

   αn×βn×γn=(αβγ)n\alpha^n \times \beta^n \times \gamma^n = (\alpha \beta \gamma)^nαn×βn×γn=(αβγ)n

9. What is αn×βn×γn×δn\alpha^n \times \beta^n \times \gamma^n \times \delta^nαn×βn×γn×δn ?

   αn×βn×γn×δn=(αβγδ)n\alpha^n \times \beta^n \times \gamma^n \times \delta^n = (\alpha \beta \gamma \delta)^nαn×βn×γn×δn=(αβγδ)n

10. What is α2 + β2\alpha^2 \, + \, \beta^2α2+β2 ?

    α2 + β2=(α + β)2−2αβ\alpha^2 \, + \, \beta^2 = (\alpha \, + \, \beta)^2 - 2 \alpha \betaα2+β2=(α+β)2−2αβ

11. What is α2 + β2 + γ2\alpha^2 \, + \, \beta^2 \, + \, \gamma^2α2+β2+γ2 ?

    α2 + β2 + γ2=(α + β + γ)2−2(αβ + βγ + γα)\alpha^2 \, + \, \beta^2 \, + \, \gamma^2 = (\alpha \, + \, \beta \, + \, \gamma)^2 - 2(\alpha \beta \, + \, \beta \gamma \, + \, \gamma \alpha)α2+β2+γ2=(α+β+γ)2−2(αβ+βγ+γα)

12. What is α2 + β2 + γ2 + δ2\alpha^2 \, + \, \beta^2 \, + \, \gamma^2 \, + \, \delta^2α2+β2+γ2+δ2 ?

    α2 + β2 + γ2 + δ2=(α + β + γ + δ)2−2(αβ + αγ + αδ + βγ + βδ + γδ)\alpha^2 \, + \, \beta^2 \, + \, \gamma^2 \, + \, \delta^2 = (\alpha \, + \, \beta \, + \, \gamma \, + \, \delta)^2 - 2(\alpha \beta \, + \, \alpha \gamma \, + \, \alpha \delta \, + \, \beta \gamma \, + \, \beta \d…

13. What is α3 + β3\alpha^3 \, + \, \beta^3α3+β3 ?

    α3 + β3=(α + β)3−3αβ(α + β)\alpha^3 \, + \, \beta^3 = (\alpha \, +\, \beta)^3 - 3 \alpha \beta (\alpha \, + \, \beta)α3+β3=(α+β)3−3αβ(α+β)

14. What is α3 + β3 + γ3\alpha^3 \, + \, \beta^3 \, + \, \gamma^3α3+β3+γ3 ?

    α3 + β3 + γ3=(α + β + γ)3−3(α + β + γ) (αβ + βγ + γα)+3αβγ\alpha^3 \, + \, \beta^3 \, + \, \gamma^3 = (\alpha \, + \, \beta \, + \, \gamma)^3 - 3(\alpha \, + \, \beta \, + \, \gamma) \, (\alpha \beta \, + \, \beta \gamma \, + \, \gamma \alpha) + 3 \alpha \beta \gammaα3+β3+γ3=(α+β+γ)3−3(α+β+γ)(αβ+βγ+γα)+3αβγ