Polynomials can be added, subtracted, multiplied, and divided using rules based on their terms and exponents. Addition and subtraction combine like terms, multiplication distributes each term across the other polynomial, and division uses either a common factor or polynomial long division.
What a polynomial is
A polynomial is an expression made from terms that are added or subtracted. A term can be a number, a variable raised to a nonnegative whole-number exponent, or a product of numbers and variables with such exponents. For example, 4x² − 3x + 7 is a polynomial: its terms are 4x², −3x, and 7.
The number multiplying a variable term is its coefficient. In 4x², the coefficient is 4. A term without a variable, such as 7, is a constant. The degree of a nonzero polynomial in one variable is the greatest exponent with a nonzero coefficient; the degree of 4x² − 3x + 7 is 2.
Expressions such as 1/x and √x are not polynomials in x because they use a negative exponent or a fractional exponent when rewritten as powers. This distinction matters because the usual polynomial operations preserve polynomial form for addition, subtraction, and multiplication, but not necessarily for division.
Adding polynomials
To add polynomials, group like terms and add their coefficients. Like terms have exactly the same variables raised to the same powers. A term in x² can combine with another x² term, but not with a term in x or a constant.
For example, add (3x² + 2x − 5) and (x² − 4x + 8). Group the matching terms: (3x² + x²) + (2x − 4x) + (−5 + 8). The result is 4x² − 2x + 3.
Writing terms in descending powers before combining them makes the matching structure easier to see. A missing power can be treated as having coefficient zero. In the example, neither polynomial has an x term with a different power to confuse with x²; keeping each power in its own group prevents accidental combinations.
Subtracting polynomials
Subtraction follows the same like-term rule, but the minus sign applies to every term in the polynomial being subtracted. A reliable method is to change the sign of each term inside the second set of parentheses, then combine like terms.
Consider (5x² + 3x − 2) − (2x² − x + 6). Removing the parentheses correctly gives 5x² + 3x − 2 − 2x² + x − 6. Combining like terms produces 3x² + 4x − 8.
The common error is to subtract only the first term inside the parentheses and leave the others unchanged. Distributing the negative sign first avoids that mistake. You can check the result by substituting a convenient value for x into both the original expression and the simplified one; they should give the same value.
Multiplying polynomials
Multiplication uses the distributive property: multiply every term in one polynomial by every term in the other, then combine like terms. When multiplying powers of the same variable, add the exponents. For example, x² times x is x³.
For (x + 3)(x + 2), multiply each term in the first parentheses by both terms in the second: x·x + x·2 + 3·x + 3·2. That gives x² + 2x + 3x + 6, which simplifies to x² + 5x + 6. Each pair of terms must be included; skipping a product changes the result.
For larger expressions, a grid can keep the products organized. Put the terms of one polynomial along the top and the terms of the other down the side, then fill each cell with the product of its row and column terms. Add the cell results and combine like terms. The grid is a bookkeeping aid; it expresses the same distributive property.
Polynomial multiplication is closed: multiplying two polynomials produces another polynomial. The degree of a product of nonzero polynomials is the sum of their degrees. For instance, a degree-two polynomial multiplied by a degree-one polynomial has degree three, unless the product is zero.
Dividing polynomials
Polynomial division has more than one method. If every term in the numerator shares a factor with the denominator, factor and cancel that common factor. Otherwise, polynomial long division can find a quotient and a remainder, much as ordinary long division does with whole numbers.
For example, (x² − 9)/(x − 3) can be simplified by factoring the numerator: x² − 9 = (x − 3)(x + 3). Canceling the shared factor gives x + 3, provided x is not 3, because the original denominator would be zero there. The simplified expression therefore agrees with the original only on the original expression’s allowed domain.
For a division that does not cancel evenly, divide the leading term of the current dividend by the leading term of the divisor. Multiply the divisor by that result, subtract, and repeat with the new expression until the remainder has lower degree than the divisor. The outcome is described by the identity:
Dividend = Divisor × Quotient + Remainder.
For example, dividing x² + 1 by x + 1 gives quotient x − 1 and remainder 2, because (x + 1)(x − 1) + 2 = x² + 1. The result can also be written as x − 1 + 2/(x + 1). Since the remainder is nonzero, the quotient alone is not equal to the original fraction.
When the result is not a polynomial
Addition, subtraction, and multiplication of polynomials always produce polynomials. Division is different: dividing one polynomial by another can produce a polynomial, a polynomial plus a fractional remainder, or an undefined expression at values that make the divisor zero.
For example, (x² + 2x)/x simplifies to x + 2 when x is not zero. Although x + 2 is a polynomial, the original fraction is undefined at x = 0. Canceling a factor does not restore a value that was excluded by the original denominator.
In general, a polynomial divided by another polynomial is a polynomial exactly when the divisor divides the numerator with zero remainder. Otherwise, the quotient is a rational expression, meaning a ratio of polynomials. Division by the zero polynomial is undefined.
How to choose a method and check your work
Identify the operation first, then choose a method that fits the expression. These habits help whether you are simplifying a short exercise or solving a longer algebra problem:
- Addition: align terms with matching powers and add their coefficients.
- Subtraction: distribute the negative sign across every term in the second polynomial before combining.
- Multiplication: make sure every term in one factor is multiplied by every term in the other.
- Division: check for common factors first; use long division when factoring does not give an immediate simplification.
- Checking: expand products, verify a division with the dividend identity, or substitute a value allowed by the original expression.
Keep the expression’s domain in mind during division. Any value that makes an original denominator zero is excluded, even if a factor later cancels. Also keep signs attached to their terms: the coefficient of −4x is −4, not 4.
Common misconceptions
Like terms are determined by their variable parts, not by whether their coefficients look similar. The terms 2x² and −7x² are like terms, but 2x² and −7x are not. Their different exponents mean they represent different kinds of terms.
Exponents also follow different rules in different operations. When multiplying powers with the same base, add exponents: x²·x³ = x⁵. When adding like terms, do not add exponents: x² + x² = 2x². These expressions describe different operations.
Finally, a nonzero remainder in polynomial division cannot be ignored. It may be written as a fraction over the divisor, but it remains part of the exact answer. If a problem asks for a polynomial quotient, check whether the division is intended to have zero remainder or whether a remainder is expected.
Frequently Asked Questions
Can you add polynomials with different degrees?
Yes. Add the coefficients of matching powers and keep terms with no match. For example, adding x² + 1 to 2x gives x² + 2x + 1.
Why do you change every sign when subtracting parentheses?
Subtracting a grouped expression means multiplying every term in it by −1. Distributing that negative sign changes the sign of each term before the terms are combined.
Does dividing polynomials always give a polynomial?
No. Division may leave a nonzero remainder, which creates a fractional part, and the expression is undefined wherever its original denominator is zero.
How do you know whether polynomial division is exact?
It is exact when the remainder is zero. You can check by multiplying the divisor by the quotient and confirming that the product equals the dividend.
Can a polynomial contain a negative exponent?
No. In a polynomial, variable exponents are nonnegative whole numbers. A negative exponent represents a reciprocal, so the expression is not a polynomial in that variable.
Summary
Polynomial addition and subtraction combine like terms; multiplication distributes across all term pairs; and division uses factoring or long division, while keeping track of remainders and excluded values. Matching exponents correctly, handling signs carefully, and checking the result are the key habits for all four operations.