One letter and its powers

The idea

Algebraic expressions involving one variable and its powers are called one-variable (univariate) polynomials, or simply polynomials.

Count the letters

Raju's total was 4x + 5y + 3. It uses two letters. A wire 20 cm long is bent into rectangles of length x. The area is 10x − x².

Which of these two expressions uses only one letter?

• 4x + 5y + 3
• 10x − x²
• Both of them

One or two?

Look at each expression. How many different letters does it use?

Polynomials

Algebraic expressions involving one variable and its powers are called one-variable polynomials, also called univariate polynomials, or simply polynomials.

The one letter can be x, y or z. The expression may use its powers too: x, x², x³ and so on.

• Expression | The one variable
• 4x | x
• x² + 1 | x
• 2y − 5 | y
• 5y³ + y² + 2y − 1 | y
• 3z + 7 | z

Notes

An algebraic expression in one variable and its powers is a one-variable (univariate) polynomial, or simply a polynomial.

Check yourself

Which of these is a polynomial in one variable?

How many different variables does 200l + 160w + 50lw use?

Answer: 2

The letters are l and w: 2 variables.

"Univariate" means…

Is 10x − x² a polynomial in one variable?

• 4x + 5y + 3. It uses two letters, x and y.
• 3z + 7 — correct. Yes. Only the letter z is used.
• 200l + 160w + 50lw. It uses two letters, l and w.
• having two variables. That would be "bivariate". Univariate means one.
• having no variable. A univariate polynomial has one variable.
• having one variable — correct. Yes. "Uni" means one, so a univariate polynomial has one variable.
• Yes. It uses only x and its power x². — correct. Right. The area of the wire rectangles is a polynomial in x.
• No. It has a power of x.. Powers of the one variable are allowed. That is what a polynomial is.
• No. It has a minus sign.. A minus sign does not matter. Only the number of letters does.