GED Math: Polynomials

A fraction is undefined when its denominator equals zero, because division by zero has no meaning. To find where an expression is undefined, set the denominator equal to 0 and solve. Watch for denominators that become zero after a substitution, such as y − x when x = y. Even if the numerator is also zero, the expression is still undefined.

1. Parts of a Polynomial

A polynomial is an expression made of one or more terms (poly means “many”). Each term can combine a coefficient (the number in front of a variable, which can be negative or a fraction; if you see none, it is 1), variables (letters), exponents (whole numbers 0, 1, 2, …), and constants (plain numbers that never change). The degree of a one-variable polynomial is its largest exponent. Polynomials are named by their number of terms: a monomial has one term, a binomial has two, and a trinomial has three. You may divide by a constant (4x/12 is fine) or put a number under a radical (√10 is fine), but the items in the table below are not allowed.

2. Combining Like Terms

You can only combine like terms: terms with exactly the same variables raised to exactly the same powers. 8x³ and 4x³ are like; 5y and 3y are like; any two plain numbers are like. But 3a and 5b are unlike (different variables), 9x and 7x² are unlike (different powers), and 2xy and 7x are unlike (one has an extra variable). To simplify, add or subtract only the coefficients of like terms, keeping the variable part. Treat a minus sign in front of a term as part of its coefficient. A tidy answer lists terms from the highest degree to the lowest. A polynomial with no like terms is already simplified.

3. Adding Polynomials

To add polynomials, remove the parentheses (a plus sign in front changes nothing) and combine like terms across both polynomials. If a polynomial needs simplifying on its own first, do that, then combine. The process is identical whether you have one variable or several; just be careful to combine only terms with the same variables and powers. For example, 2xy + 7xy = 9xy, but 2xy + 7x cannot be combined.

4. Subtracting Polynomials

The only difference from adding is the minus sign between the parentheses. Distribute that minus sign to every term of the second polynomial, which reverses each of its signs (+ becomes − and − becomes +). Then drop the parentheses and combine like terms. A common error is to flip only the first term. Remember that subtracting a negative is the same as adding a positive.

5. Multiplying Monomials and Monomial-Binomial Products

To multiply monomials, multiply the coefficients and then add the exponents of matching variables (a variable with no exponent has an exponent of 1). So (2ab)(4b) = 8ab², because b¹ × b¹ = b². Variables with no partner just carry through. To multiply a monomial by a binomial, use the distributive property: multiply the monomial by each term and add the results, keeping track of negative signs.

6. Multiplying Two Binomials with FOIL

To multiply two binomials, multiply First terms, Outer terms, Inner terms, and Last terms, then add the four products and combine like terms. FOIL is really the distributive property used twice: every term in the first binomial multiplies every term in the second. Keep signs with their terms, and add exponents when you multiply matching variables.

7. Dividing Polynomials

To simplify a polynomial written as a fraction, divide the numerator and the denominator by a common factor. Often you must first factor the common term out of the numerator. If the numerator has several terms, you may also split it into separate fractions and divide each term by the denominator. The common factor can be a number, a variable with a coefficient, or even a whole expression such as (x + 7). Always divide every term, not just the first.

8. Undefined Expressions

A fraction is undefined when its denominator equals zero, because division by zero has no meaning. To find where an expression is undefined, set the denominator equal to 0 and solve. Watch for denominators that become zero after a substitution, such as y − x when x = y. Even if the numerator is also zero, the expression is still undefined.

Common Mistakes to Avoid

• Calling an expression a polynomial when it has a variable in a denominator, a negative or fractional exponent, or a variable under a radical.
• Combining unlike terms, such as 2xy + 7x or 9x + 7x².
• Forgetting that a lone variable has coefficient 1 and exponent 1.
• When subtracting, reversing the sign of only the first term of the second polynomial instead of every term.
• Adding exponents when you add terms. Exponents only add when you multiply (x² × x³ = x⁵), and they stay the same when you add like terms (3x² + 4x² = 7x²).
• Skipping the middle term when squaring a binomial: (x + y)² is not x² + y².
• Stopping FOIL before combining like terms.
• Dividing only one term of the numerator by the denominator.
• Forgetting that a denominator equal to zero makes an expression undefined.

Frequently Asked Questions