Polynomials and Operations

If you can follow a recipe, you can work with polynomials. Really. A polynomial is just an expression with multiple ingredients - some terms with variables, maybe some plain numbers, all combined together. You have already been working with simpler versions of these throughout algebra. Now we are going to see how these multi-term expressions behave when you add, subtract, and multiply them.

Core Concepts

What is a Polynomial?

A polynomial is an expression made up of terms that are added or subtracted together. Each term consists of a coefficient (a number) multiplied by a variable raised to a whole number power. That is the key requirement: whole number exponents only. No fractions in the exponent, no negative exponents, no variables in the denominator.

Here are some examples of polynomials:

• $3x^2 + 2x - 5$
• $7y^4 - y + 12$
• $x^3$
• $8$

And here are some expressions that are not polynomials:

• $\frac{3}{x}$ (variable in denominator - same as $3x^{-1}$)
• $2x^{1/2} + 5$ (fractional exponent)
• $4^x$ (variable in the exponent)

Think of it this way: if you can write the expression using only addition, subtraction, multiplication, and whole number exponents, it is a polynomial.

Classifying Polynomials by Number of Terms

Polynomials get special names based on how many terms they have:

Name	Number of Terms	Examples
Monomial	1 term	$5x^3$, $-7$, $2xy$
Binomial	2 terms	$x + 3$, $4x^2 - 9$
Trinomial	3 terms	$x^2 + 2x + 1$, $3a^2 - a + 7$
Polynomial	4 or more terms	$x^4 - 3x^3 + 2x^2 - x + 5$

For polynomials with four or more terms, we simply call them polynomials (no special name). The prefixes “mono-”, “bi-”, and “tri-” mean one, two, and three - the same prefixes you see in words like “bicycle” (two wheels) and “triangle” (three angles).

Degree of a Polynomial

The degree of a polynomial is the highest exponent that appears on any variable. This tells you a lot about how the polynomial behaves.

• $4x^3 + 2x - 7$ has degree 3 (the highest exponent is 3)
• $x^2 + 5x + 6$ has degree 2
• $9x + 1$ has degree 1
• $15$ has degree 0 (constants are degree 0)

For terms with multiple variables, add the exponents together. For example, in $3x^2y^3$, the degree of that term is $2 + 3 = 5$.

Why does degree matter? The degree tells you what kind of polynomial you have. Degree 1 is linear (straight line when graphed), degree 2 is quadratic (parabola), degree 3 is cubic, and so on. Each degree has its own characteristic shape and behavior.

Standard Form

A polynomial is in standard form when its terms are arranged from highest degree to lowest degree, left to right.

$$6x^4 - 3x^3 + x^2 - 5x + 2$$

This is standard form. The exponents decrease as you read from left to right: 4, 3, 2, 1, 0.

The leading coefficient is the coefficient of the first term (the term with the highest degree). In the polynomial above, the leading coefficient is 6.

Writing polynomials in standard form makes them easier to compare, add, and work with. It is like organizing a bookshelf - things are easier to find when they are in order.

Adding Polynomials

Adding polynomials is straightforward: combine like terms. Like terms have the same variable raised to the same power.

To add $(3x^2 + 2x - 1) + (x^2 - 4x + 5)$:

1. Remove the parentheses (addition does not change any signs)
2. Group like terms
3. Combine

$$3x^2 + 2x - 1 + x^2 - 4x + 5$$ $$= (3x^2 + x^2) + (2x - 4x) + (-1 + 5)$$ $$= 4x^2 - 2x + 4$$

Think of it like combining ingredients: 3 cups of flour plus 1 cup of flour gives you 4 cups of flour. You can only combine things that are the same “type.”

Subtracting Polynomials

Subtracting polynomials requires one extra step: distribute the negative sign to every term in the polynomial being subtracted.

To subtract $(5x + 3) - (2x - 7)$:

1. Distribute the negative sign to the second polynomial
2. Combine like terms

$$5x + 3 - 2x + 7$$

Notice that $-(2x - 7)$ becomes $-2x + 7$. The negative flips both signs.

$$= (5x - 2x) + (3 + 7)$$ $$= 3x + 10$$

Common mistake alert: Forgetting to distribute the negative to every term in the second polynomial. Always write out the step where you distribute the negative - it will save you from errors.

Multiplying Polynomials

When multiplying polynomials, every term in the first polynomial must be multiplied by every term in the second polynomial. This is the distributive property applied multiple times.

Multiplying a Monomial by a Polynomial:

$$3x(2x^2 + 5x - 4)$$ $$= 3x \cdot 2x^2 + 3x \cdot 5x + 3x \cdot (-4)$$ $$= 6x^3 + 15x^2 - 12x$$

Multiplying Two Binomials (FOIL):

For two binomials specifically, there is a helpful pattern called FOIL:

• First terms
• Outer terms
• Inner terms
• Last terms

For $(x + 3)(x - 5)$:

• First: $x \cdot x = x^2$
• Outer: $x \cdot (-5) = -5x$
• Inner: $3 \cdot x = 3x$
• Last: $3 \cdot (-5) = -15$

Combine: $x^2 - 5x + 3x - 15 = x^2 - 2x - 15$

Multiplying Larger Polynomials:

For polynomials with more terms, use the distributive property systematically. Multiply each term of the first polynomial by each term of the second, then combine like terms.

Special Products

Some multiplication patterns appear so often that they are worth memorizing:

Perfect Square of a Sum: $$(a + b)^2 = a^2 + 2ab + b^2$$

Perfect Square of a Difference: $$(a - b)^2 = a^2 - 2ab + b^2$$

Difference of Squares: $$(a + b)(a - b) = a^2 - b^2$$

These patterns save time and reduce errors once you recognize them. Notice that $(a + b)^2$ is not $a^2 + b^2$ - a common mistake. You must include the middle term $2ab$.

Notation and Terminology

Term	Meaning	Example
Polynomial	Sum of terms with variables raised to whole number powers	$3x^2 + 2x - 5$
Monomial	One term	$5x^3$
Binomial	Two terms	$x + 3$
Trinomial	Three terms	$x^2 + 2x + 1$
Degree	Highest exponent	Degree of $4x^3 + x$ is 3
Leading coefficient	Coefficient of highest-degree term	In $4x^3 + x$, it is 4
Like terms	Terms with same variable and exponent	$3x^2$ and $5x^2$
Standard form	Terms ordered from highest to lowest degree	$2x^3 - x^2 + 5x - 1$
Constant term	The term without a variable	In $x^2 + 3x + 7$, it is 7

Examples

Key Properties and Rules

Adding and Subtracting Polynomials

• Only combine like terms - terms must have the same variable(s) with the same exponent(s)
• When subtracting, distribute the negative sign to every term in the polynomial being subtracted
• The result will have degree equal to the highest degree among the polynomials being combined

Multiplying Polynomials

• Every term multiplies every term - use the distributive property completely
• When multiplying terms, add the exponents: $x^2 \cdot x^3 = x^{2+3} = x^5$
• When multiplying terms, multiply the coefficients: $3x^2 \cdot 4x^3 = 12x^5$
• The degree of the product equals the sum of the degrees of the factors

Special Products to Memorize

Pattern	Formula	Example
Perfect Square (Sum)	$(a+b)^2 = a^2 + 2ab + b^2$	$(x+3)^2 = x^2 + 6x + 9$
Perfect Square (Diff)	$(a-b)^2 = a^2 - 2ab + b^2$	$(x-4)^2 = x^2 - 8x + 16$
Difference of Squares	$(a+b)(a-b) = a^2 - b^2$	$(x+5)(x-5) = x^2 - 25$

Standard Form Checklist

A polynomial is in standard form when:

1. Terms are ordered from highest degree to lowest degree
2. Like terms have been combined
3. The leading coefficient is identified (coefficient of the highest-degree term)

Real-World Applications

Area Calculations

Suppose a rectangular garden has length $(x + 5)$ feet and width $(x + 2)$ feet. The area is:

$$A = (x + 5)(x + 2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10 \text{ square feet}$$

If you later decide $x = 3$ feet, the area is $9 + 21 + 10 = 40$ square feet.

Projectile Motion

The height of a ball thrown upward is often modeled by a polynomial:

$$h(t) = -16t^2 + 40t + 5$$

This trinomial tells you the height $h$ in feet at time $t$ seconds. The $-16t^2$ accounts for gravity, $40t$ represents the initial upward velocity, and $5$ is the starting height.

Revenue and Profit Functions

A business might model revenue as:

$$R(x) = 50x - 0.5x^2$$

where $x$ is the number of units sold. This polynomial captures that revenue increases with sales, but at some point, selling more requires lowering the price (the $-0.5x^2$ term).

If costs are $C(x) = 10x + 200$, profit is:

$$P(x) = R(x) - C(x) = (50x - 0.5x^2) - (10x + 200) = -0.5x^2 + 40x - 200$$

Compound Interest (Simplified)

If you invest $P$ dollars and it grows by a factor of $(1 + r)$ each period, after 2 periods you have:

$$P(1 + r)^2 = P(1 + 2r + r^2)$$

The polynomial $(1 + 2r + r^2)$ shows how your money grows: the original amount, plus two rounds of simple interest ($2r$), plus “interest on interest” ($r^2$).

Geometric Formulas

Many geometry formulas are polynomials. The surface area of a cube with side length $s$:

$$SA = 6s^2$$

The volume of a rectangular prism with dimensions $x$, $(x + 1)$, and $(x + 2)$:

$$V = x(x+1)(x+2) = x(x^2 + 3x + 2) = x^3 + 3x^2 + 2x$$

Self-Test Problems

Summary

• A polynomial is an expression made of terms with variables raised to whole number powers, combined by addition and subtraction. Think of each term as an ingredient in a recipe.
• Polynomials are classified by their number of terms: monomial (1), binomial (2), trinomial (3), or just polynomial (4+).
• The degree is the highest exponent in the polynomial. It determines the polynomial’s fundamental behavior and shape when graphed.
• Standard form arranges terms from highest to lowest degree. The leading coefficient is the coefficient of the first term.
• To add polynomials, combine like terms. To subtract, distribute the negative sign first, then combine like terms.
• To multiply polynomials, use the distributive property: every term in the first polynomial multiplies every term in the second. For two binomials, FOIL (First, Outer, Inner, Last) is a helpful pattern.
• Special products are patterns worth memorizing: $(a + b)^2 = a^2 + 2ab + b^2$, $(a - b)^2 = a^2 - 2ab + b^2$, and $(a + b)(a - b) = a^2 - b^2$.
• Polynomials appear throughout mathematics and real-world applications: calculating areas, modeling motion, analyzing business functions, and describing geometric relationships.