2.1 Expressions & Polynomial Arithmetic
Key Takeaways
The distributive property a(b + c) = ab + ac drives expanding, combining like terms, and GCF factoring throughout Algebra I.
Like terms share identical variable parts and exponents; only their coefficients add or subtract.
Polynomial multiplication uses the distributive property; FOIL is just the four products of two binomials.
Exponent laws multiply by adding exponents, divide by subtracting, and raise a power to a power by multiplying exponents.
Regents items reward interpreting parts of an expression (coefficient, factor, term) in context, not just computing.
Why Expressions Anchor the Algebra Strand
The Algebra strand carries 48%-61% of Regents credits, and almost every Algebra question begins by reading or rewriting an expression. An expression is a combination of numbers, variables, and operations with no equals sign, such as 3x^2 - 5x + 2. Knowing how to evaluate it, simplify it, and interpret its parts is the gateway skill for equations, functions, and quadratics later in the exam.
Vocabulary the Regents uses precisely:
- A term is a number, variable, or product separated by + or - (in 3x^2 - 5x + 2 there are three terms).
- A coefficient is the number multiplying a variable (the coefficient of -5x is -5).
- A factor is one of the quantities being multiplied (in 4(x + 3), both 4 and (x + 3) are factors).
The Distributive Property and Combining Like Terms
The distributive property says a(b + c) = ab + ac. It is the single most reused rule in the course. To simplify 4(2x - 3) + 5x, first distribute, then combine like terms.
Worked example: 4(2x - 3) + 5x = 8x - 12 + 5x = (8x + 5x) - 12 = 13x - 12.
Like terms have the exact same variable part. So 8x and 5x combine, but 8x and 8x^2 do not, because x and x^2 are different. Watch subtraction of a grouped quantity: the minus sign distributes to every term inside.
Worked example: 7y - (3y - 8) = 7y - 3y + 8 = 4y + 8. A common trap is writing 7y - 3y - 8; the second sign must flip because you are subtracting the whole quantity.
When several groups appear, distribute each one before combining. Simplify 2(3x + 1) - (x - 4): distribute to get 6x + 2 - x + 4, then combine to 5x + 6. A number standing alone, like +2 and +4 here, is a constant term and combines only with other constants, never with x terms. Many Regents items simply ask which choice is equivalent to a given expression, so simplifying accurately is the whole task.
Adding, Subtracting, and Multiplying Polynomials
A polynomial is a sum of terms with whole-number exponents. To add polynomials, combine like terms. To subtract, distribute the negative sign first.
Subtraction example: (3x^2 + 2x - 1) - (x^2 - 4x + 6). Distribute the minus: 3x^2 + 2x - 1 - x^2 + 4x - 6. Combine: 2x^2 + 6x - 7.
To multiply two binomials, distribute every term of the first across the second (FOIL is just these four products).
Multiplication example: (x + 5)(x - 3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15. The middle term comes from adding the two cross-products (-3x and +5x), a step students often drop.
Squaring a binomial is the same process: (x - 4)^2 means (x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16. It is not x^2 + 16; the cross-terms matter. Multiplying a monomial across a polynomial also uses distribution: 2x(x^2 - 3x + 5) = 2x^3 - 6x^2 + 10x, where each term gets the 2x and the exponents add.
GCF Factoring and the Exponent Laws
Factoring out the greatest common factor (GCF) reverses the distributive property. In 6x^3 + 9x^2, the GCF is 3x^2, so 6x^3 + 9x^2 = 3x^2(2x + 3). Check by re-distributing: 3x^2 times 2x is 6x^3 and 3x^2 times 3 is 9x^2.
The properties of exponents appear constantly:
| Rule | Statement | Example |
|---|---|---|
| Product | x^a times x^b = x^(a+b) | x^4 times x^3 = x^7 |
| Quotient | x^a divided by x^b = x^(a-b) | x^5 / x^2 = x^3 |
| Power of a power | (x^a)^b = x^(ab) | (x^2)^3 = x^6 |
| Power of a product | (xy)^a = x^a y^a | (2x)^3 = 8x^3 |
| Zero exponent | x^0 = 1 (x not 0) | 7^0 = 1 |
Notice (2x)^3 = 2^3 x^3 = 8x^3: the coefficient is also raised to the power, a frequent slip.
Evaluating and Interpreting Expressions in Context
To evaluate, substitute values and follow the order of operations. Evaluate 2x^2 - 3x for x = -4: 2(-4)^2 - 3(-4) = 2(16) + 12 = 32 + 12 = 44. Square before multiplying, and remember (-4)^2 = 16, not -16.
Regents interpretation items ask what a part of an expression means. If a phone repair shop charges C = 25 + 40h, then 25 is the fixed service fee in dollars and 40 is the hourly labor rate in dollars per hour. In a factored revenue model R = p(120 - 4p), the factor p is the price and (120 - 4p) models the quantity sold at that price. Being able to name coefficient, factor, and term in plain language is worth as many credits as computing.
A second common interpretation task involves growth. In the savings model 500(1.04)^t, the 500 is the starting amount in dollars, the 1.04 is the yearly growth factor (a 4% increase), and t counts years. Reading 1.04 as "100% kept plus 4% added" connects expression structure to its real meaning, exactly the reasoning Regents constructed-response prompts ask students to explain.
Simplify the expression 5(2x - 4) - 3(x + 2).
7x - 26
13x - 14
7x - 14
10x - 18
Which expression is equivalent to (x + 6)(x - 2)?
x^2 - 12
x^2 + 8x - 12
x^2 + 4x - 12
x^2 - 4x - 12
Sections you finish are checked off in the contents.