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Polynomials : Algebraic Division and Complex Roots

The polynomial:

4x^4 - 6x^3 + 4x^2 - 3x + 1 has the real roots at x = 1 and x = 1/2 and two complex conjugate roots.

(a) by the process of algebraic division and then solving a quadratic equation, find the complex roots.

(b) Write down all the factors of the 4th degree polynomial

4x^4 - 6x^3 + 4x^2 - 3x + 1

(c) Find the gradient of the curve y = 4x^4 - 6x^3 + 4x^2 - 3x + 1 at the point where x=2.
(note - x is the letter x)

Solution Preview

The polynomial:

4x^4 - 6x^3 + 4x^2 - 3x + 1 has the real roots at x = 1 and x = 1/2 and two complex conjugate roots.

(a) By the process of algebraic division and then solving a quadratic equation, find the complex roots.
Since x=1 and x=1/2 are roots, (x-1)(2x-1)=2x^2-3x+1 divides the given polynomial ...

Solution Summary

Algebraic division of a polynomial and complex roots are found.

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