Polynomial arithmetic
The idea
Polynomials are expressions built from variables and constants using only addition, subtraction, and multiplication — things like 2x³ − 11x² + 22x − 15. You already manipulate algebraic expressions and exponent rules; polynomial arithmetic organizes that into a system that behaves remarkably like the integers: add, subtract, or multiply two polynomials and you always get another polynomial. That closure is why polynomials are the workhorses of algebra, modeling everything from areas to profit curves.
Adding and subtracting is collecting like terms — terms with the same variable part. Multiplying is the distributive law applied thoroughly: every term of the first factor must multiply every term of the second, and the degrees add, so a degree-1 polynomial times a degree-2 polynomial gives degree 3. The classic error is freelancing with shortcuts like claiming (a + b)² equals a² + b², which skips the cross terms; careful expansion gives a² + 2ab + b². A quick guard against slips is to evaluate both the original factors and your expansion at an easy value like x = 1 and confirm they match.
Worked example
Expand (2x − 3)(x² − 4x + 5), combine like terms, and verify the result by evaluating at x = 1.
- Distribute 2x across the trinomial: 2x × x² = 2x³, then 2x × (−4x) = −8x², then 2x × 5 = 10x.
- Distribute −3 the same way: −3 × x² = −3x², then −3 × (−4x) = +12x, then −3 × 5 = −15. Keeping each sign attached to its term prevents the most common errors.
- Collect like terms: the x² terms give −8x² − 3x² = −11x², and the x terms give 10x + 12x = 22x, so the product is 2x³ − 11x² + 22x − 15.
- Verify at x = 1: the factors give (2 − 3)(1 − 4 + 5) = (−1)(2) = −2, and the expansion gives 2 − 11 + 22 − 15 = −2. The match is strong evidence the expansion is correct.
Answer. The product is 2x³ − 11x² + 22x − 15.
Check your understanding
- Why must the product of a degree-1 and a degree-2 polynomial always have degree 3, and what does that predict about the term count before combining?
- How is multiplying two polynomials like multiplying two multi-digit numbers place by place?
- Why does checking at a single value like x = 1 catch most expansion errors yet still fall short of a guarantee?
- What exactly goes missing when someone writes (a + b)² = a² + b², and why does the error vanish when a or b is zero?
Build the foundations first
Polynomial arithmetic builds on these concepts. If any feel shaky, start there.