13.1 Polynomial Operations
Key Takeaways
- A polynomial is a sum of terms of the form $ax^n$ with nonnegative integer exponents; degree is the highest exponent, and the leading coefficient multiplies that highest-power term
- Add or subtract by combining like terms only—same variable and same exponent; redistribute a leading minus when subtracting a whole polynomial
- Multiply by the distributive property (or FOIL for two binomials); every term of the first factor multiplies every term of the second
- After multiplying, always combine like terms and write the product in standard form (descending powers)
- On PERT Math, sign errors and incomplete distribution are the most common traps—expand fully before simplifying
13.1 Polynomial Operations
Quick Answer: A polynomial is an expression built from constants and variables raised to whole-number powers, combined by addition and subtraction. Add/subtract by combining like terms. Multiply by distributing every term of one factor across every term of the other (use FOIL for two binomials), then combine like terms and write the result in standard form (highest power first).
PERT Mathematics lists polynomials—simplify, add, subtract, multiply, and divide—as a core algebra competency. Placement items rarely ask for abstract definitions alone; they ask you to produce an equivalent simplified polynomial and match it to an answer choice. This section builds the vocabulary and the three main operations you will use again when you factor, solve quadratics, and simplify rational expressions.
Vocabulary You Need on Test Day
| Term | Meaning | Example in $4x^3 - 7x^2 + 5x - 9$ |
|---|---|---|
| Term | Product of a number and powers of variables, or a constant alone | $4x^3$, $-7x^2$, $5x$, $-9$ |
| Coefficient | Numerical factor of a term | $4$, $-7$, $5$, $-9$ |
| Degree of a term | Sum of exponents on variables in that term | $3$, $2$, $1$, $0$ |
| Degree of a polynomial | Highest degree among its terms | $3$ |
| Leading term | Term of highest degree (in standard form, written first) | $4x^3$ |
| Leading coefficient | Coefficient of the leading term | $4$ |
| Constant term | Term with no variable (degree 0) | $-9$ |
| Monomial / binomial / trinomial | Polynomial with 1 / 2 / 3 terms | $3x^2$; $x+2$; $x^2+5x+6$ |
Standard form means descending powers of the variable: $3x^2 - 5x + 1$, not $1 - 5x + 3x^2$. Answer keys almost always use standard form, so rewrite before you compare options.
Like terms share the exact same variable factors with the same exponents. $3x^2$ and $-8x^2$ are like; $3x^2$ and $3x$ are not; $2xy$ and $2x$ are not. Only like terms can be combined by adding or subtracting their coefficients.
Adding Polynomials
To add, drop parentheses (addition does not change signs) and combine like terms.
Worked Example 1: Add Two Trinomials
Problem: Simplify $(5x^2 - 3x + 7) + (2x^2 + 9x - 4)$.
Step 1 — Remove parentheses:
Step 2 — Group like terms (optional but safe):
Step 3 — Combine coefficients:
Check: Substitute $x = 1$. Original sum: $(5-3+7) + (2+9-4) = 9 + 7 = 16$. Result: $7+6+3 = 16$. Match.
Subtracting Polynomials
Subtraction means add the opposite of every term in the second polynomial. Distribute the leading minus carefully—missing a sign change is the classic trap.
Worked Example 2: Subtract with Sign Changes
Problem: Simplify $(4x^3 - 2x + 5) - (x^3 + 6x^2 - 3x + 1)$.
Step 1 — Distribute the minus: change every sign inside the second parentheses:
Step 2 — Combine like terms:
Common trap: Writing $- (x^3 + 6x^2 - 3x + 1)$ as $-x^3 + 6x^2 - 3x + 1$ or forgetting to flip the $-3x$ to $+3x$. Every term flips: $-x^3$, $-6x^2$, $+3x$, $-1$.
Multiplying: Distributive Property
For a monomial times a polynomial, distribute the monomial to every term:
When multiplying powers of the same base, add exponents: $x^m \cdot x^n = x^{m+n}$. Multiply numerical coefficients separately.
Worked Example 3: Monomial Times Trinomial
Problem: Multiply $3x^2(2x^3 - 5x + 4)$.
Step 1 — Distribute:
Step 2 — Multiply each piece:
(No like terms to combine.) Written in standard form, this is already done.
Why it matters for PERT: Later division-by-monomial items reverse this process. If you cannot expand cleanly, you will not reverse cleanly either.
FOIL for Two Binomials
When both factors are binomials, FOIL organizes the four required products:
| Letter | Meaning | For $(a+b)(c+d)$ |
|---|---|---|
| F | First | $a \cdot c$ |
| O | Outer | $a \cdot d$ |
| I | Inner | $b \cdot c$ |
| L | Last | $b \cdot d$ |
Then combine the middle terms when they are like.
Worked Example 4: Classic FOIL
Problem: Multiply $(x + 5)(x - 3)$.
F: $x \cdot x = x^2$
O: $x \cdot (-3) = -3x$
I: $5 \cdot x = 5x$
L: $5 \cdot (-3) = -15$
Combine: $x^2 + (-3x + 5x) - 15 = x^2 + 2x - 15$
Check with FOIL reverse (factoring): numbers that multiply to $-15$ and add to $2$ are $5$ and $-3$. Yes.
Worked Example 5: FOIL with Coefficients
Problem: Multiply $(2x - 7)(3x + 4)$.
F: $2x \cdot 3x = 6x^2$
O: $2x \cdot 4 = 8x$
I: $-7 \cdot 3x = -21x$
L: $-7 \cdot 4 = -28$
Combine: $6x^2 + (8x - 21x) - 28 = 6x^2 - 13x - 28$
Sign check: Outer is positive; Inner is negative; Last is negative because negative times positive. Students who treat $-7$ as unsigned often get $+28$ or wrong middle signs.
Worked Example 6: Special Products (Useful Shortcuts)
Memorize patterns that appear often:
| Pattern | Expanded form |
|---|---|
| $(a+b)^2$ | $a^2 + 2ab + b^2$ |
| $(a-b)^2$ | $a^2 - 2ab + b^2$ |
| $(a+b)(a-b)$ | $a^2 - b^2$ (difference of squares) |
Problem: Expand $(3x - 2)^2$.
Using $(a-b)^2$ with $a = 3x$, $b = 2$:
Trap: Writing $(3x - 2)^2 = 9x^2 - 4$ (forgetting the middle term). Squaring a binomial is not the same as squaring each term alone.
Multiplying Beyond Two Binomials
For a binomial times a trinomial (or larger), still distribute every term of the first factor across every term of the second, then combine.
Worked Example 7: Binomial × Trinomial
Problem: Multiply $(x + 2)(x^2 - 3x + 5)$.
Distribute $x$: $x(x^2 - 3x + 5) = x^3 - 3x^2 + 5x$
Distribute $2$: $2(x^2 - 3x + 5) = 2x^2 - 6x + 10$
Add:
Degree check: Degree of product should equal sum of degrees of factors ($1 + 2 = 3$). Leading term $x \cdot x^2 = x^3$. Good sanity check on PERT.
Putting Operations Together
Some items nest operations: multiply first (or distribute), then add/subtract, then combine.
Worked Example 8: Mixed Operations
Problem: Simplify $2(x - 4) - (x^2 - 3x + 1) + (x + 5)(x - 5)$.
Step 1 — Expand each piece:
Step 2 — Combine everything:
Step 3 — Like terms: $(-x^2 + x^2) + (2x + 3x) + (-8 - 1 - 25) = 5x - 34$
The $x^2$ terms cancel—a frequent intended result on multi-step simplify items.
Strategy Checklist for PERT Polynomial Items
- Identify the operation (add, subtract, multiply, or mixed).
- If subtracting, flip all signs in the second polynomial before combining.
- If multiplying, expand fully—every term times every term—before combining.
- Combine only like terms; leave unlike terms separate.
- Write standard form and match the option that is algebraically identical (not merely similar).
- Optional check: substitute a convenient number (such as $x = 2$) into original and simplified expressions; unequal values mean an algebra error.
Master these mechanics before moving to division and factoring. Every later polynomial skill on PERT Math is reverse engineering of what you practiced here: combining, distributing, and tracking signs with care.
Simplify: $(3x^2 - 5x + 2) + (x^2 + 4x - 9)$. Which expression is equivalent?
Simplify: $(6x^2 - 3x + 1) - (2x^2 + 5x - 4)$. What is the result?
Multiply: $(x + 6)(x - 2)$. Which product is correct?
Expand $(2x - 5)^2$. Which expression is correct?