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Higher-degree equations

Polynomial equations

Japanese school year: Math II

What you learn

You will learn how to solve polynomial equations of degree three or higher using the factor theorem and special factoring techniques. These skills are used to find graph intersections and solve geometric volume problems. Prior mastery of the factor theorem and quadratic equations will help significantly.

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Key points

To solve a higher-degree polynomial equation, find a number α\alpha where P(α)=0P(\alpha) = 0 to factor out (x−α)(x - \alpha) and reduce the degree of the equation.

P(α)=0  ⟹  P(x)=(x−α)Q(x)=0P(\alpha) = 0 \implies P(x) = (x - \alpha)Q(x) = 0

Rational solutions to integer-coefficient polynomial equations can only come from ±(factor of constant term a0)/(factor of leading coefficient an)\pm(\text{factor of constant term } a_0) / (\text{factor of leading coefficient } a_n). Use this to narrow down candidates.

x=±pq(p∣a0,  q∣an)x = \pm \frac{p}{q} \quad (p \mid a_0, \; q \mid a_n)

For biquadratic equations containing only x4x^4 and x2x^2, solve by substituting X=x2X = x^2 or by rewriting into a difference of squares to factor.

x4+ax2+b=(x2+k)2−(mx)2=0x^4 + ax^2 + b = (x^2 + k)^2 - (mx)^2 = 0

In the field of complex numbers, counting repeated roots individually, an nn-th degree polynomial equation has exactly nn solutions.

Choose a set to practice.