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The remainder and factor theorems
The remainder theorem P(a) as the remainder on division by x − a, the factor theorem, factorising cubics and solving polynomial equations.
The theorems
Factorise $P(x) = x^3 - 4x^2 + x + 6$.
Solve $x^3 - 4x^2 + x + 6 = 0$.
$(x + 1)(x - 2)(x - 3) = 0$.
Remainder when $x^3 + 2x - 5$ is divided by $x - 1$?
$P(1) = 1 + 2 - 5$.
Is $x - 2$ a factor of $x^3 - 4x^2 + x + 6$?
$P(2) = 8 - 16 + 2 + 6 = 0$.
Is $x + 2$ a factor of $x^3 - 4x^2 + x + 6$?
$P(-2) = -8 - 16 - 2 + 6 = -20$.
$x - 1$ is a factor of $x^3 + kx - 5$. Find $k$.
$1 + k - 5 = 0$.
Remainder when $2x^3 - x + 3$ is divided by $x - 2$?
$16 - 2 + 3$.
Solve $x^3 + 2x^2 - x - 2 = 0$.
$x^2(x + 2) - (x + 2) = (x + 2)(x - 1)(x + 1)$.
Remainder when $x^2 + 4x$ is divided by $x + 1$?
$P(-1) = 1 - 4$.
New numbers every time. Keep going until you get five in a row.