10.1 Add and Subtract Polynomials

Polynomials

A polynomial is a math expression that combines variables and numbers using addition, subtraction, and multiplication. Learning to recognize, add, and subtract polynomials is the foundation for working with more complex algebra later on.

Types of Polynomials

Polynomials are named by how many terms they have. A term is a piece of the expression separated by a $+$ or $-$ sign.

• Monomial: a polynomial with just one term ($3x$, $-5y^2$, $7$)
• Binomial: a polynomial with exactly two terms ($2x + 3$, $x^2 - 4y$)
• Trinomial: a polynomial with exactly three terms ($x^2 + 2x + 1$, $3x^2 - 5x + 2$)

Any expression with four or more terms is simply called a polynomial.

Components of Polynomials

Before you start adding and subtracting, make sure you can identify each part of a polynomial:

• Variable: a letter representing an unknown quantity ($x$, $y$, $z$)
• Coefficient: the number multiplied by the variable in a term. In $7x^2$, the coefficient is $7$.
• Constant term: a term with no variable attached, like the $+1$ in $x^2 + 2x + 1$
• Terms: the individual pieces of a polynomial, separated by $+$ or $-$ signs

Degree of a Polynomial

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

• $3x^2$ has degree 2
• $2x^3 + 3x^2 - 5x + 1$ has degree 3, because $x^3$ is the highest power
• A constant like $7$ has degree 0

The degree tells you a lot about how the polynomial behaves when graphed. Higher-degree polynomials produce more complex curves with more turning points.

Addition and Subtraction of Polynomials

The core idea here is combining like terms. Like terms have the same variable raised to the same power. You can only combine terms that match.

For example, $3x^2$ and $5x^2$ are like terms, but $3x^2$ and $5x$ are not.

Adding polynomials:

1. Remove the parentheses (signs stay the same).
2. Group like terms together.
3. Add the coefficients of each group.

Subtracting polynomials:

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

The most common mistake with subtraction is forgetting to distribute the negative to every term inside the second set of parentheses. Double-check each sign after you remove the parentheses.

Simplifying Polynomial Expressions

Simplifying just means combining all the like terms into one clean expression. Follow these steps:

1. Identify all like terms (same variable, same exponent).
2. Add or subtract their coefficients.
3. Write the result in standard form, which means terms go in descending order of degree (highest exponent first).

Evaluating Polynomials

To evaluate a polynomial, plug in the given value for the variable and then simplify using order of operations.

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

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

2. Exponent first: $2(4) - 3(2) + 1$

3. Multiply: $8 - 6 + 1$

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

A quick tip: always handle the exponent before multiplying by the coefficient. A common error is to multiply $2 \times 2$ first and then square, which gives you the wrong answer.