Long Division & Rearranging

Section 1 of 2

Polynomial Long Division

Dividing one polynomial by another is the same dance as the long division you learned with whole numbers — set up the bus-stop and chip away at the dividend until it’s gone.

The four steps

Divide. Multiply. Subtract. Bring down. Repeat. Divide the leading terms, multiply the divisor by that, subtract, and carry on.

Worked example — a walk-through

Divide $x^2 + 9x + 18$ by $(x + 3)$.

$x^2 \div x = x$; $x(x+3) = x^2 + 3x$; subtract → $6x + 18$

$6x \div x = 6$; $6(x+3) = 6x + 18$; subtract → $0$

$x + 6$

Worked example — another

Divide $x^2 + 11x + 30$ by $(x + 5)$.

$x^2 \div x = x$; $x(x+5) = x^2 + 5x$; subtract → $6x + 30$

$6x \div x = 6$; $6(x+5) = 6x + 30$; subtract → $0$

$x + 6$

You try

Divide $x^2 + 11x + 24$ by $(x + 8)$.

$x^2 \div x = x$. Multiply $x(x+8)$, subtract, then divide again.

$x(x+8) = x^2 + 8x$; subtract → $3x + 24$

$3x \div x = 3$; $3(x+8) = 3x + 24$; subtract → $0$

$x + 3$

Section 2 of 2

Rearranging Formulae

Rearranging a formula means getting a chosen letter alone on one side. The rule is the same as solving any equation.

The rule

Whatever you do to one side, do to the other. When the letter is on both sides: multiply out, gather all the $x$ terms on one side, factorise the $x$ out, then divide.

Worked example — write x in terms of the others

Given $px + q = r$, write $x$ in terms of $p, q, r$.

$px = r - q$

$x = \dfrac{r - q}{p}$

Worked example — with a single bracket

Given $a(x + 1) = b$, write $x$ in terms of $a$ and $b$.

$ax + a = b$

$ax = b - a$

$x = \dfrac{b - a}{a}$

Worked example — letters on both sides

Given $p(x + 1) = q(x + 2)$, write $x$ in terms of $p$ and $q$.

$px + p = qx + 2q$

$px - qx = 2q - p$

$x(p - q) = 2q - p$

$x = \dfrac{2q - p}{p - q}$

You try

Given $a(x - 3) = b(x + 5)$, write $x$ in terms of $a$ and $b$.

Multiply out both brackets, gather the $x$ terms on one side, factorise $x$, then divide.

$ax - 3a = bx + 5b$

$ax - bx = 5b + 3a$

$x(a - b) = 5b + 3a$

$x = \dfrac{5b + 3a}{a - b}$

That’s Class 5.

Long division (divide, multiply, subtract, bring down) and rearranging formulae to make a letter the subject. Class 6: solving linear equations and inequalities.