Evaluating Polynomials

To evaluate a polynomial means to find its value at a specific number, by substituting that number in for the variable and simplifying. It's the same idea as evaluating any algebraic expression, applied to a polynomial's multiple terms.

For 2x² − 3x + 1 at x = 4: first substitute 4 for every x: 2(4)² − 3(4) + 1. Then simplify with order of operations — exponents first: 2(16) − 3(4) + 1. Then multiply: 32 − 12 + 1. Finally add and subtract: 32 − 12 + 1 = 21.

Substituting is the easy part; getting the arithmetic right afterward depends entirely on following order of operations correctly — exponents before multiplication, multiplication before addition and subtraction. A single skipped step, like multiplying before squaring, gives a wrong answer even with a correct substitution. This builds on evaluating algebraic expressions.

Substituting a negative number needs extra care with parentheses: evaluating x² at x = −3 means (−3)² = 9, not −3² = −9. Always wrap a negative substitution in parentheses before simplifying, a habit that carries over from understanding what a polynomial is and identifying its terms and coefficients in the first place.

Related lessons

• Algebra