Algebraic Expressions

This topic is the vocabulary layer under the rest of algebra. The questions look easy, and they are, provided you never combine two things that are not the same kind of thing.

The vocabulary

Take the expression 7x squared minus 4x plus 9.

• Terms are the pieces separated by plus and minus signs: 7x squared, -4x, and 9. There are three.
• Coefficient is the number multiplying a variable: 7 and -4. The sign belongs to the coefficient. The second term's coefficient is negative 4, not 4, and forgetting that causes more errors than anything else in the topic.
• Constant is the term with no variable: 9.
• Variable is the letter: x.

Two conventions the test relies on:

A variable with no visible coefficient has a coefficient of 1. x means 1x, and -x means -1x.

A variable with no visible exponent has an exponent of 1.

Like terms

Same variable, same exponent. Both conditions, every time.

To combine, add the coefficients and leave the variable part alone.

7x + 3y - 2x + 5y

The x terms: 7x minus 2x is 5x. The y terms: 3y plus 5y is 8y.

Result: 5x + 8y. It does not simplify further, because x and y are not like terms. An answer of 13xy is the standard error and it is always a choice.

Distributing first

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Most questions make you clear a parenthesis before anything can combine.

Simplify 4(2x - 3) - 2(x + 5)

• First: 4 times 2x is 8x; 4 times -3 is -12.
• Second: the -2 multiplies both terms, giving -2x and -10.
• Now: 8x - 12 - 2x - 10.
• Combine: 8x minus 2x is 6x; -12 minus 10 is -22.

Result: 6x - 22.

The trap is the second parenthesis. Distributing only the 2 and keeping the minus sign in front gives 8x - 12 - 2x + 10, or 6x - 2, which is a choice. A minus sign in front of a parenthesis is a factor of -1 and it hits every term inside.

That is why -(x - 7) is -x + 7.

Evaluating at a value

Substitute and compute, respecting order of operations.

Evaluate 3a squared - 2b when a = -2 and b = 5.

Write the substitution with parentheses.

3(-2) squared minus 2(5).

• Exponent first: (-2) squared is 4. Note the parentheses did their job here; the square of negative 2 is positive 4.
• 3 times 4 is 12.
• 2 times 5 is 10.
• 12 minus 10 is 2.

Substituting without parentheses gives 3 x -2 squared, which invites reading it as the negative of 3 times 4, and produces -12 - 10, or -22. That is a choice, and the parentheses are what prevent it.

Function notation

f(x) is just an expression with a name. "f(x) = 2x + 1" means "the rule f takes a number, doubles it and adds 1", and f(3) means put 3 in for x: 2(3) + 1 = 7.

A function inside a function is worked from the inside out. For f(x) = 3x - 5 and g(x) = x + 2, f(g(3)) means find g(3) first - that is 5 - and then f(5), which is 15 - 5 = 10.

Translating words into expressions

The test asks you to write an expression from a phrase.

Two of these decide most questions.

"Less than" reverses the order. Five less than a number is n - 5, never 5 - n. This is the most commonly missed translation in algebra and the reversed version is always a choice.

"Twice the sum" needs parentheses; "twice a number plus five" does not. 2(n + 5) is not 2n + 5. Listen for whether the word "sum" or "difference" comes before the quantity, which signals that the whole quantity is being operated on.

Multiplying expressions

Beyond distributing one term, the test asks for products of two binomials, which belongs to the factoring lesson. What it asks here is simpler:

Multiply the coefficients, add the exponents of matching variables.

(3x squared)(4x cubed) is 12x to the fifth.

(2ab)(5a) is 10a squared b.

What you can skip

Across the 53 questions on this topic:

• Domain, range and graphs of functions never appear. Function questions are substitution, as above.
• Polynomial division and rational expressions never appear in this topic.

Where people lose points

Combining unlike terms. 3x plus 4y is not 7xy.

Adding exponents when adding terms. 3x plus 5x is 8x. The exponents only add when the terms are multiplied.

Dropping a sign on distribution, especially with a minus in front of a parenthesis.

Losing a negative on substitution, which parentheses prevent.

Reversing "less than".

Forgetting the invisible 1 in front of a lone variable, so that x + 5x comes out as 5x rather than 6x.

Work one in under a minute

Simplify 5(x - 2) - (3x - 8)

First: 5x - 10.

Second: the minus in front is -1 across both terms, giving -3x + 8.

Combine: 5x minus 3x is 2x; -10 plus 8 is -2.

Result: 2x - 2.

Where this leads

Everything. Solving equations is combining like terms on both sides of an equals sign, and factoring is distributing in reverse.