Algebraic Expressions

An algebraic expression is built from terms joined by $+$ or $-$ signs:

$3x^2 - 4xy + 7y - 2.$

Each term has:

• A coefficient (the number in front, e.g. 3 in $3x^2$).
• Letters with powers (e.g. $x^2$, $xy$, $y$).
• A constant term may have no letters (e.g. $-2$).

LIKE TERMS have the SAME letters with the SAME powers. Coefficient differences are fine.

• $3x$ and $7x$ → like.
• $5xy$ and $-2xy$ → like.
• $4x^2$ and $4x$ → NOT like ($x^2 \ne x$).
• $3xy$ and $3yx$ → like ($xy = yx$).

You can ADD or SUBTRACT like terms by combining their coefficients:

$5x + 3x = 8x.$ $7x^2y - 4x^2y = 3x^2y.$

You CANNOT combine UNLIKE terms. $3x + 5y$ stays as $3x + 5y$.

Worked example. Simplify $4a + 7b - 2a + 5b - 3$.

• Group like: $(4a - 2a) + (7b + 5b) - 3$.
• Combine: $2a + 12b - 3$.

Single bracket: multiply everything inside by what's outside.

$a(b + c) = ab + ac.$

Worked example. Expand $3(2x - 5y + 4)$.

• $3 \times 2x = 6x$.
• $3 \times (-5y) = -15y$.
• $3 \times 4 = 12$.
• Result: $6x - 15y + 12$.

Two brackets (FOIL). $(a + b)(c + d) = ac + ad + bc + bd$.

The acronym FOIL (First, Outer, Inner, Last) helps you remember to multiply every term in the first bracket by every term in the second.

Worked example. Expand $(x + 3)(x + 5)$.

• F: $x \times x = x^2$.
• O: $x \times 5 = 5x$.
• I: $3 \times x = 3x$.
• L: $3 \times 5 = 15$.
• Sum: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$.

Worked example (Extended). Expand $(2x - 3)(x + 4)$.

• F: $2x \times x = 2x^2$.
• O: $2x \times 4 = 8x$.
• I: $-3 \times x = -3x$.
• L: $-3 \times 4 = -12$.
• Sum: $2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$.

Three special identities to memorise:

• $(a+b)^2 = a^2 + 2ab + b^2$.
• $(a-b)^2 = a^2 - 2ab + b^2$.
• $(a+b)(a-b) = a^2 - b^2$ (difference of squares).

These show up CONSTANTLY in factorisation and algebra problems.

Substitution means replacing each letter with its given value, then evaluating.

Worked example. Find the value of $3x^2 - 2x + 5$ when $x = -4$.

ALWAYS use brackets around the substituted value:

• $3(-4)^2 - 2(-4) + 5$.
• BIDMAS: $(-4)^2 = 16$.
• $3 \times 16 - 2 \times (-4) + 5 = 48 + 8 + 5 = 61$.

(Without brackets, $-4^2$ would be $-16$ — wrong by sign.)

Worked example. Find the value of $\dfrac{a^2 - b^2}{2a + b}$ when $a = 5$ and $b = 3$.

• Top: $5^2 - 3^2 = 25 - 9 = 16$.
• Bottom: $2(5) + 3 = 13$.
• Result: $\dfrac{16}{13}$.

Multi-step. Sometimes you need to substitute into one formula then into another:

Volume of a cone: $V = \tfrac{1}{3}\pi r^2 h$. Find $V$ when $r = 3$ and $h = 5$.

• $V = \tfrac{1}{3} \times \pi \times 3^2 \times 5 = \tfrac{1}{3} \times \pi \times 9 \times 5 = 15\pi \approx 47.1$.

A simple rule: at the start of any substitution problem, REWRITE the expression with brackets around each substituted value. This single habit removes most sign errors.