Roots of Polynomials Q (1 Viewer)

[chevyd00]

Could anyone please explain what to do in this question and why:

Use the fourth degree equation ax⁴ + bx³ + cx² +dx + e = 0 whose roots are alpha, beta, gamma and delta to verify that the equation P(x/m)=0 has roots m times those of P(x)=0.

Thank you!

 

[jyu]

Could anyone please explain what to do in this question and why:

Use the fourth degree equation ax⁴ + bx³ + cx² +dx + e = 0 whose roots are alpha, beta, gamma and delta to verify that the equation P(x/m)=0 has roots m times those of P(x)=0.

Thank you!

The polynomial P(x/m) is the dilation of P(x) by a factor of m in the x-direction, .: all roots of P(x/m)=0 are m times those of P(x)=0.

 

[chevyd00]

I've expanded P(x/m) to get ax⁴ + bx³m + cx²m² + dxm³ + em⁴ = 0, and I know that the next thing I need to do is use the roots m(alpha), m(beta), m(gamma) and m(delta), but what next? What exactly am I verifying?

 

[braintic]

I've expanded P(x/m) to get ax⁴ + bx³m + cx²m² + dxm³ + em⁴ = 0, and I know that the next thing I need to do is use the roots m(alpha), m(beta), m(gamma) and m(delta), but what next? What exactly am I verifying?

P(x) = 0 has roots (solutions for x): x = α, β, γ, δ.

P(x/m) = 0 has solutions for x/m: x/m = α, β, γ, δ

ie. x = mα, mβ, mγ, mδ

The actual equation is irrelevant.