Factor theorem $f(a)=0\iff(x-a)\text{ is a factor}$note the SIGN: the factor $(x-3)$ pairs with $f(3)$
Remainder theorem $f(a)=\text{remainder on dividing by }(x-a)$a remainder of zero IS the factor theorem
Linear factor $(ax-b)$ $f\!\left(\dfrac ba\right)=0$e.g. $(2x-1)$ is a factor when $f(\tfrac12)=0$
Find an unknown $\text{substitute and solve}$if $(x-2)$ is a factor, set $f(2)=0$ and solve for the letter
Factorise a cubic $f(x)=(x-a)(\text{quadratic})$find one root, divide out, then factorise what is left
Test a factor · Is $(x-2)$ a factor of $f(x)=x^{3}-3x^{2}+4$?
$f(2)=8-12+4$
$=0$
yes, $(x-2)$ is a factor
Find an unknown · $(x+1)$ is a factor of $x^{3}+kx-6$. Find $k$.
$(x+1)$ means $f(-1)=0$
$-1-k-6=0$
$k=-7$
Factorise fully · Factorise $x^{3}-7x+6$ given $(x-1)$ is a factor.
divide: $x^{3}-7x+6=(x-1)(x^{2}+x-6)$
factorise the quadratic: $(x+3)(x-2)$
$=(x-1)(x+3)(x-2)$
Polynomial: an expression of whole-number powers of $x$, e.g. a cubic.
Root: a value of $x$ making the expression zero - it corresponds to a factor.
Remainder: what is left after dividing; the remainder theorem finds it without dividing.
Cubic: a polynomial whose highest power is $x^{3}$; it has up to three roots.
✗ Testing $f(-2)$ for the factor $(x-2)$
✓ the factor $(x-2)$ pairs with $f(2)$.
✗ Concluding a factor because $f(a)$ is small
✓ it must be exactly ZERO.
✗ Stopping after finding one factor of a cubic
✓ "factorise fully" means all three linear factors where they exist.
✗ Testing $x=5$ for $x^{3}-7x+6$
✓ only factors of the constant $6$ can be whole-number roots.
• The sign flips: the factor $(x-3)$ corresponds to substituting $x=3$, and $(x+1)$ to $x=-1$.
• Once you have one factor, division reduces the cubic to a quadratic - which you already know how to factorise.
• To find a first root by trial, test the factors of the CONSTANT term: they are the only whole-number candidates.
• List the factors of the constant term and test them in order - the root is nearly always small.
• After dividing, always try to factorise the quadratic before writing the final answer.
• Check by expanding your factorisation back to the original.
Factor Theorem