Arithmetic with Polynomials and Rational Expressions
Polynomials
Find the degree of a polynomial, evaluate it at a value and recognize its zeroes.
A polynomial is an expression built from a letter using only adding, subtracting and multiplying — so 3x² - 5x + 2 is one, and so is 7 on its own. What is not allowed is dividing by the letter or putting it under a root sign. Writing it as p(x) turns it into a machine: feed a number in, get a number out.
Degree, value, zero
The degree is the highest power of x that appears. The value p(k) is what comes out when k goes in. And a zero of the polynomial is a value of x that makes the whole thing come out as 0 — which is what an equation is really asking you for.
Worked example
For p(x) = x² - 5x + 6, find p(2).
Replace every x with 2: p(2) = 2² - 5 × 2 + 6.
Every x, not just the first — a missed one is the commonest slip here.
Try it together
For p(x) = x² - 4x + 1, find p(3).
Put 3 in wherever an x appears, then work through it a piece at a time.
1.The first term is x². With x = 3, what is 3 × 3?
Watch out
The degree counts powers of the letter, not how many terms there are. 2x³ + x has two terms and degree 3; 7 has one term and degree 0, because 7 is 7x⁰.
Have a go
Have a go on your own: what is the degree of 7x³ - 2x + 5?
Hint: Look for the highest power of x anywhere in it.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.