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).
Which expression is equivalent to (x + 6)(x - 2)?