Polynomial Expressions and Equations
A polynomial equation of degree n has at most n real solutions, because its highest power controls how many times its graph can cross the x-axis. Combine polynomials by matching like terms, multiply by distributing every term, and factor to find where they equal zero.
The degree of a polynomial is its highest exponent, and that number is a ceiling on how many real solutions the equation can have when set equal to zero. A quadratic (degree 2) has at most 2 real roots, a cubic (degree 3) has at most 3, and so on. This comes from the graph: each real root is a place the curve crosses the x-axis, and a degree-n curve can only bend enough times to cross n times at most. Some roots can repeat or go complex instead, which is why you often see fewer real solutions than the degree suggests.
Adding and subtracting polynomials just means combining like terms: same variable, same exponent, nothing else. Multiplying means distributing every term in one factor across every term in the other, then collecting like powers again — miss a term here and the whole answer shifts. Factoring by grouping handles four-term polynomials: split them into two pairs, pull out the common factor from each pair, and if a common binomial appears in both, factor that out too. This turns a sum into a product, and a product equal to zero means at least one factor is zero — that's how you solve.
This topic connects directly to quadratics and rational expressions elsewhere in the algebra set: factoring a polynomial is the same move you use to simplify a rational expression or solve a quadratic by factoring. It also sets up the remainder theorem, which asks what's left over when a polynomial doesn't factor evenly — a natural next step once grouping and multiplying feel automatic.
Key terms
- Degree
- The highest exponent on the variable in a polynomial, which caps its number of real roots.
- Like terms
- Terms with the same variable raised to the same power, the only terms you can add or subtract directly.
- Factoring by grouping
- Splitting a four-term polynomial into two pairs, factoring each pair, then factoring out their shared binomial.
- Root
- A value of the variable that makes the polynomial equal zero, also called a solution or zero.
Watch it explained
- How it worksHow do you add, subtract and multiply polynomials without dropping a term on the way?
- How it worksHow does factoring by grouping crack a four-term polynomial open?
- Worked exampleFind the product of (x - 3) and (x2 + 2x - 5).
- Worked exampleFind every real solution of x3 - 4x = 0 by factoring it completely.
- Going furtherWhat happens to the degree of a polynomial when you multiply it by another polynomial?
Check yourself
End of topic.