6.1 Add and Subtract Polynomials

Types of Polynomials and Their Degrees

Types of polynomials

A polynomial is an algebraic expression made up of variables and coefficients, combined using addition, subtraction, and multiplication. Variables are typically letters like $x$, $y$, or $z$, and coefficients are the numbers multiplied by those variables.

Polynomials are classified by how many terms they have:

• Monomial: a single term, such as $3x$, $-5y^2$, or $7$
• Binomial: exactly two terms, such as $2x + 3$ or $x^2 - 4y$
• Trinomial: exactly three terms, such as $x^2 + 2x + 1$ or $3x^3 - 2x^2 + 5x$

Terms are separated by addition or subtraction signs. So in $x^2 + 2x + 1$, there are three terms: $x^2$, $2x$, and $1$.

Degree of polynomials

The degree of a polynomial is the highest exponent on any variable in the expression.

• For a monomial, just look at the exponent. The degree of $3x^2$ is 2.
• For a polynomial with multiple terms, find the term with the largest exponent. The degree of $2x^3 + 3x^2 - 4x + 1$ is 3, because $x^3$ has the highest power.
• A constant like $7$ has degree 0 (since $7 = 7x^0$).

The degree of a polynomial affects its shape when graphed. Higher-degree polynomials produce more complex curves with more possible turning points, but for this unit, the main skill is identifying the degree correctly.

Adding and Subtracting Polynomials

Addition and subtraction of polynomials

Adding and subtracting polynomials comes down to one core idea: combine like terms. Like terms are terms that have the same variable raised to the same exponent. For example, $3x^2$ and $-5x^2$ are like terms, but $3x^2$ and $3x$ are not.

To add polynomials:

1. Write the polynomials next to each other (drop the parentheses).
2. Group like terms together.
3. Add the coefficients of each group.

To subtract polynomials:

1. Distribute the negative sign to every term in the polynomial being subtracted (flip each sign).
2. Group like terms together.
3. Combine the coefficients.

The most common mistake here is forgetting to distribute the negative sign to every term in the second polynomial. Double-check each sign after you remove the parentheses.

Properties of polynomial operations

Two familiar properties apply when you're rearranging and grouping terms:

• Commutative property: Order doesn't matter for addition or multiplication. $a + b = b + a$
• Associative property: Grouping doesn't matter for addition or multiplication. $(a + b) + c = a + (b + c)$

These properties are why you can freely rearrange terms to group like terms together when adding or subtracting polynomials.

Evaluation of polynomials

To evaluate a polynomial, substitute the given value for each variable, then simplify using order of operations.

Evaluate $2x^2 - 3x + 1$ when $x = 2$:

1. Substitute 2 for every $x$: $2(2)^2 - 3(2) + 1$

2. Handle the exponent first: $2(4) - 3(2) + 1$

3. Multiply: $8 - 6 + 1$

4. Add and subtract left to right: $3$

The value of the polynomial when $x = 2$ is $3$.

Real-world applications of polynomials

Polynomials show up in problems involving area, volume, revenue, and cost. Here's a typical example:

A rectangular garden has a length that is 3 meters longer than its width. If the width is $x$ meters, the area is $x(x + 3) = x^2 + 3x$.

• The width is $x$ meters.
• The length is $x + 3$ meters.
• The area polynomial $x^2 + 3x$ lets you calculate the area for any width. If $x = 5$, the area is $5^2 + 3(5) = 25 + 15 = 40$ square meters.