Quick questions on Factor and remainder theorems: VCE Math Methods Unit 3

Test small integers (especially $\pm 1$ and factors of the constant term). Substitute into $P$ and check for zero.

Once you have a root $a$, divide $P(x)$ by $(x - a)$ to get a quadratic quotient. Use polynomial long division or synthetic division.

Use any of the standard quadratic methods: factor, complete the square, or the quadratic formula. If the discriminant is negative, the quadratic is irreducible over the reals and you stop at $(x - a) (\text{irreducible quadratic})$.

Factorise $P(x) = x^3 + x^2 - 8x - 12$ over the rationals.

For a polynomial with integer coefficients, the rational root theorem says that any rational root $p/q$ in lowest terms must have $p$ dividing the constant term and $q$ dividing the leading coefficient. So for $P(x) = 2x^3 - 3x^2 + 4x - 6$, the candidates are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 1/2, \pm 3/2$.

Only test rational numbers $p/q$ where $p$ divides the constant and $q$ divides the leading coefficient.

Each subtraction step changes signs. Write out the division carefully.

"Fully factorise" means continue until each factor is linear or irreducible quadratic.

Dividing by $(x - a)$ uses $P(a)$, not $P(-a)$. Dividing by $(x + 3)$ uses $P(-3)$.

When the quadratic quotient shares a root with the original linear factor, you get a repeated root. Always check.