Polynomials & Cubic Equations

At a glance 11 formulas

1. 1Remainder theorem$P(x) \div (x - a) \text{ leaves } P(a)$$P(x) \div (ax - b) \text{ leaves } P\left(\tfrac{b}{a}\right)$
2. 2Factor theorem$(x - a) \text{ is a factor of } P(x) \iff P(a) = 0$
3. 3Division algorithm$P(x) = D(x)\,Q(x) + R(x)$$D \text{ divisor},\ Q \text{ quotient},\ R \text{ remainder},\ \deg R < \deg D$
4. 4Remainder by a quadratic$P(x) \div (x - a)(x - b) \text{ leaves } px + q$$pa + q = P(a),\ pb + q = P(b)$
5. 5Cubic: sum and product$r_1 + r_2 + r_3 = -\dfrac{b}{a},\ r_1r_2r_3 = -\dfrac{d}{a}$$r_1r_2 + r_2r_3 + r_3r_1 = \dfrac{c}{a};\ r_1, r_2, r_3 \text{ the roots of } ax^3 + bx^2 + cx + d = 0$
6. 6Sum of reciprocals$\dfrac{1}{r_1} + \dfrac{1}{r_2} + \dfrac{1}{r_3} = -\dfrac{c}{d}$$r_1, r_2, r_3 \text{ roots of } ax^3 + bx^2 + cx + d = 0$
7. 7Factored form$P(k) = a(k - r_1)(k - r_2)(k - r_3)$$r_1, r_2, r_3 \text{ the roots of } P,\ a \text{ its leading coefficient}$
8. 8Roots in APIf a cubic's roots are in AP, write them $m - d$, $m$, $m + d$: the sum of roots gives $m$ at once, and $m$ is itself a root.
9. 9Values on a lineIf a cubic $P$ has $P(1), P(2), P(3)$ on a line $mx + c$, then $P(x) - (mx + c)$ is 0 at 1, 2 and 3, so it is $(x - 1)(x - 2)(x - 3)$ times $P$'s leading coefficient.
10. 10Surd inputTo evaluate $P(x)$ at $x = a + \sqrt{b}$, use $(x - a)^2 = b$ to cut the powers down before substituting.
11. 11Compare coefficientsWhen two polynomials are equal for every $x$, match the coefficients of each power: $x^3$ with $x^3$, $x^2$ with $x^2$, down to the constants.