Algebraic expressions

An algebraic expression is notation where letters stand beside numbers: $3x + 5$, $2(a - b)$, $\frac{x}{2} - 1$. A letter stands for a number — one we do not know yet, or one that is allowed to change. That way a single expression describes a whole family of situations instead of one case at a time.

A ticket costs 12 units and a seat reservation 5. The cost of buying $x$ tickets with one reservation is:

This one line answers the question for any number of tickets — just substitute a number for $x$.

Term, coefficient, variable

An expression is built from terms — the parts separated by plus and minus signs. In $12x + 5$ the terms are $12x$ and $5$.

• variable — the letter standing for a number ($x$),
• coefficient — the number in front of the variable ($12$),
• constant term — a term with no variable ($5$).

Writing $12x$ means multiplication: $12 \cdot x$. The multiplication sign between a number and a letter is dropped, because nothing else could be meant.

Like terms and how to collect them

Like terms are terms with the same letter raised to the same power. $3x$ and $7x$ are like terms; $3x$ and $3x^2$ are not, and neither are $3x$ and $3y$.

Like terms can be merged into one — that is collecting like terms. Only the coefficients are added; the letter stays as it is:

This is not a new rule but the distributive law read right to left: $3x + 2x = (3 + 2)x = 5x$. Three apples and two apples make five apples — not "five apples squared".

Collecting terms does not "work anything out". It writes the same object more briefly — and a shorter expression is easier to evaluate and harder to get wrong.

The distributive law

Multiplying a number by a sum means multiplying it by every term:

The area of a rectangle shows why: a rectangle with sides $a$ and $b + c$ can be cut into two rectangles of areas $ab$ and $ac$. The same rule holds for subtraction: $a(b - c) = ab - ac$.

A minus in front of a bracket

A minus in front of a bracket is multiplication by $-1$, so every term inside flips its sign — including one that was already negative:

This is where a sign gets lost most often. When a bracket is being subtracted, expand first and collect terms afterwards.

Multiplying two binomials

A binomial is a sum of two terms, such as $x + 3$. Multiplying two brackets together needs no new rule — it is the same distributive law, applied twice. First treat the second bracket as a single number:

Four products are left: every term of the first bracket times every term of the second. The area of a rectangle shows it again — a rectangle with sides $a + b$ and $c + d$ cuts into four smaller ones of areas $ac$, $ad$, $bc$ and $bd$, and together they make up the whole.

The number of products is always the product of the numbers of terms: a binomial times a binomial is four multiplications, a binomial times a trinomial is six. Fewer than that means one has been missed.

Two of these products come up so often that they have been written down once and for all — $(a + b)^2$ and $(a - b)(a + b)$ are what the special products are about. The rule in this section, however, always applies, including when no formula fits.

Factoring out the common term

The distributive law can also be read right to left. A sum then turns into a product:

This is the reverse of expanding a bracket and it is called factoring out the common term. Collecting like terms was a special case of it: $ax + bx = (a + b)x$ is the same move, with $x$ as the factor taken out.

The common factor has two parts:

• the numeric part — the greatest common divisor of the coefficients,
• the letter part — the lowest power of a letter that stands in every term.

The factor has to be the greatest one available. Writing $3x^2(4x + 6)$ is arithmetically correct but only half-factored: $4$ and $6$ are still both even. A finished factorization is recognized by the terms inside the bracket having nothing left in common.

Factoring is not an exercise for its own sake. A product says things about an expression that a sum does not — when it is zero, for one — which is why cancelling an algebraic fraction and solving an equation in factored form both start here.

The value of an expression

To evaluate an expression, substitute a number for the letter and carry out the operations in the usual order. For $8x + 6$ with $x = 4$:

Simplifying an expression never changes its value — $5x + 8 + 3x - 2$ with $x = 4$ also gives $38$. That is the simplest way to check a rearrangement: substitute the same number into both forms and compare.

The exercises below ask for an expression. What counts is its value, not how you write it: $5x + 10$, $10 + 5x$ and $5 \cdot x + 10$ are one and the same answer. Copying the question back is not — the point is the simpler form. Where the task is to factor, simpler means a product: the answer is $6x^2(2x + 3)$, not the sum expanded back out.

Common mistakes

• Adding unlike terms — $3x + 4$ is already the answer; there is no "$7x$" to be had.
• Confusing adding with multiplying letters — $x + x = 2x$, but $x \cdot x = x^2$.
• Losing a sign after a minus in front of a bracket — $-(x - 3)$ is $-x + 3$, not $-x - 3$.
• Expanding only the first term — in $3(2x + 4)$ the three multiplies both terms, so $6x + 12$, not $6x + 4$.
• Multiplying brackets "straight across" — $(x + 3)(x - 5)$ is four products, not $x^2 - 15$.
• Taking out too small a factor — $12x^3 + 18x^2$ gives $6x^2(2x + 3)$, not $3x^2(4x + 6)$.