Understanding Polynomials and Their Terms
A polynomial is a sum of monomials—each a coefficient multiplied by a variable raised to a whole-number power. When we write 3x² + 2x + 5, we're looking at three distinct terms: 3x², 2x, and the constant 5.
The degree of a polynomial is the highest exponent appearing in any term. In 5x⁴ − 3x² + 7, the degree is 4. Polynomials with specific numbers of terms have special names: a monomial contains one term (like 7x³), a binomial has two terms (like x + 3), and a trinomial has three terms (like x² − 2x + 1).
The key insight for addition and subtraction is recognising like terms—monomials with identical variables and exponents. You can only combine 5x² with −2x², but 5x² and 5x must remain separate because their powers differ.
The Addition and Subtraction Rules
When adding two polynomials P(x) and Q(x), align terms by degree and sum their coefficients. When subtracting Q from P, distribute the negative sign through Q's coefficients, then combine like terms. If P and Q have different degrees, include all terms from both—missing terms are treated as having coefficient zero.
P(x) + Q(x) = Σ(aₙ + bₙ)xⁿ
P(x) − Q(x) = Σ(aₙ − bₙ)xⁿ
P(x), Q(x)— The two input polynomialsaₙ, bₙ— Coefficients of xⁿ in P and Q respectivelyn— The exponent (degree of each term)
Step-by-Step Method for Addition
To add two polynomials by hand:
- Write both polynomials in descending order of degree. Arrange terms from highest to lowest power.
- Align like terms vertically. Place all x⁴ terms in one column, all x³ terms in another, and so on.
- Add the coefficients of aligned terms. Ignore the variable and exponent; just add the numbers in front.
- Keep terms with zero coefficient implicit. If a term sums to zero, you can omit it from your final answer.
Example: (2x³ + 5x + 3) + (4x³ − 2x² + x − 1) = 6x³ − 2x² + 6x + 2
Step-by-Step Method for Subtraction
Subtracting one polynomial from another is nearly identical to addition, with one crucial difference:
- Distribute the negative sign through the second polynomial. Change every coefficient in Q(x) to its opposite before combining.
- Treat it as addition thereafter. Once you've flipped the signs, the process is identical to adding polynomials.
Example: (5x² + 3x + 2) − (2x² + x − 4) = (5x² + 3x + 2) + (−2x² − x + 4) = 3x² + 2x + 6
A frequent mistake is forgetting to flip the sign of every term in the second polynomial. Pay special attention to the constant term—if it's positive in Q, it becomes negative when you subtract.
Common Pitfalls and Best Practices
Avoid these frequent errors when combining polynomials:
- Forgetting the negative sign in subtraction — When subtracting, you must negate every single coefficient in the second polynomial. Forgetting the negative on the constant term or missing a middle term is a classic mistake. Always write out the flipped signs explicitly before combining.
- Mixing up different variable powers — You can only combine terms with identical variables raised to identical powers. Terms like <code>3x²</code> and <code>3x³</code> look similar but are fundamentally different and cannot be combined. Check both the variable and its exponent.
- Dropping missing terms from higher degrees — If one polynomial has degree 5 and the other has degree 3, the degree-5 and degree-4 terms must still appear in your result. Treat missing terms as having coefficient zero, not as absent.
- Sign errors with negative coefficients — When a polynomial coefficient is already negative (e.g., <code>−3x</code>) and you're subtracting, double-check: subtracting <code>−3x</code> becomes <code>+3x</code>. Write out the signs carefully to catch these.