Solving cubic and quartic equations by factoring, by synthetic division and as quadratics in form, and solving rational equations that lead to quadratics.
The strategy: factor and split
There's no simple formula for most equations of degree \(3\) or higher. Instead, get \(0\) on one side, factor, and use the zero product property on each factor.
A polynomial equation of degree \(n\) has at most\(n\) real solutions.
Factoring
Synthetic division
If nothing factors at sight, look for an integer root among the factors of the constant term. Each root \(r\) gives a factor \((x - r)\), and synthetic division finds the other factor.
Equations of the form \(x^n = k\)
Quadratic in form
An equation like \(x^4 - 13x^2 + 36 = 0\) only has the powers \(x^4\),\(x^2\) and a constant. Since \(x^4 = (x^2)^2\), it's quadratic in form: substitute \(u = x^2\).
Rational equations
A rational equation can also turn into a quadratic once you clear the denominators.
Practice
Quick check
What is the real solution of \(x^3 = -8\)?
The index is odd, so there's exactly one real cube root: \((-2)^3 = -8\).
Quick check
How many real solutions does \(x^4 + 5x^2 + 4 = 0\) have?
With \(u = x^2\):\(u = -1\) or \(u = -4\). A square can't be negative, so there are no real solutions.
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