Lesson 1: Polynomials

New Concept

A polynomial is an expression with one or more terms where variables have whole number exponents. This lesson covers how to identify polynomials, classify them by degree and terms, and perform operations like addition and subtraction.

What’s next

You're ready to start! Next up, you'll work through interactive examples on identifying, adding, and subtracting polynomials to master these core skills.

Property

A polynomial is an algebraic expression with several terms. Each term is a power of a variable (or a product of powers) with a constant coefficient. The exponents in a polynomial must be whole numbers, which means that a polynomial has no radicals containing variables, and no variables in the denominators of fractions.

Examples

• The expression $4x^3 + 2x^2 - 7x + 5$ is a polynomial.

• The expression $2 + \frac{6}{x}$ is not a polynomial because a variable appears in the denominator.

Property

In a term containing only one variable, the exponent of the variable is called the degree of the term. The degree of a polynomial in one variable is the largest exponent that appears in any term. Polynomials in one variable are usually written in descending powers of the variable.

Examples

• The polynomial $3y^2 - 2y + 2$ has a degree of 2.

• The polynomial $m - 2m^2 + m^5$ has a degree of 5.

Property

We evaluate polynomials the same way we evaluate any other algebraic expression: by substituting the given values for the variables and then following the order of operations to simplify.

Examples

• To evaluate $2x^3 - 5x + 4$ for $x = 2$, we compute $2(2)^3 - 5(2) + 4 = 2(8) - 10 + 4 = 16 - 10 + 4 = 10$.

• To evaluate $16t^3 - 6t + 20$ for $t = \frac{3}{2}$, we compute $16(\frac{3}{2})^3 - 6(\frac{3}{2}) + 20 = 16(\frac{27}{8}) - 9 + 20 = 54 - 9 + 20 = 65$.

Property

To add two polynomials we remove parentheses and combine like terms. Like terms are any terms that are exactly alike in their variable factors. The exponents on the variable factors must also match. To add like terms, we add their numerical coefficients.

Examples

• To simplify $3x^2 + 5x^2$, we add the coefficients to get $(3+5)x^2 = 8x^2$. The expression $3x^2 + 5x^3$ cannot be simplified.

• Add $(4a^3 - 2a^2 - 3a + 1) + (2a^3 + 4a - 5)$. This simplifies to $6a^3 - 2a^2 + a - 4$.

Property

If an expression in parentheses is preceded by a minus sign, we must change the sign of each term within parentheses when we remove the parentheses. This rule applies when we subtract polynomials.

Examples

• Subtract $(5x^2 + 4x - 1) - (2x^2 - 3x + 4)$. This becomes $5x^2 + 4x - 1 - 2x^2 + 3x - 4$, which simplifies to $3x^2 + 7x - 5$.

• A common mistake is not distributing the negative to all terms, such as writing $5x^2 + 4x - 1 - 2x^2 - 3x + 4$. This is incorrect.

Property

To calculate the profit it earns, a company must subtract the cost of producing goods from the revenue it earns by selling them. We can state this as a formula:

Examples

• If a company's revenue is $R = 15x$ and its cost is $C = 250 + 4x$, the profit is $P = (15x) - (250 + 4x) = 11x - 250$.

• If Revenue is $R = -0.01x^2 + 8x$ and Cost is $C = 2x + 200$, the Profit is $P = (-0.01x^2 + 8x) - (2x + 200) = -0.01x^2 + 6x - 200$.