The boss power names it

The idea

Polynomials of degree (highest power of x) 1, 2 and 3 are linear, quadratic and cubic, with general forms ax + b, ax² + bx + c and ax³ + bx² + cx + d (real coefficients, a ≠ 0).

Three polynomials

2x − 3, y² − 2 and 3x³ − 2x² + x − 1. They are called linear, quadratic and cubic.

What decides the name?

• How many terms it has
• The highest power of the variable
• The biggest coefficient

Build your own

Use + and − to choose a, b, c, d. Watch the name plate.

Degree names the polynomial

The degree of a polynomial is its highest power. Degree 1 is linear, degree 2 is quadratic, degree 3 is cubic.

The general forms, with real numbers a, b, c, d: ax + b, where a ≠ 0 ax² + bx + c, where a ≠ 0 ax³ + bx² + cx + d, where a ≠ 0

Why a ≠ 0? If a = 0, the boss term vanishes and the name drops. The word “quadratic” comes from “quadrate”, which means “square”.

Notes

The degree is the highest power. Degree 1, 2, 3 are linear, quadratic, cubic: ax + b, ax² + bx + c, ax³ + bx² + cx + d, with a ≠ 0.

Check yourself

Which of these is not a linear polynomial?

What is the degree of 5x³ − 4x² + x − √2?

Answer: 3

The powers of x are 3, 2 and 1, and the last term has no x. The highest is 3, so the degree is 3: a cubic polynomial.

In ax² + bx + c, why must a ≠ 0?

Which of these is not a polynomial?

• √3x + 5. The highest power of x is 1, so it is linear.
• y + √2. The highest power of y is 1, so it is linear.
• 2x + 5 − x² — correct. Yes! The highest power is x², so the degree is 2. It is not linear.
• (2/3)u + 1. The highest power of u is 1, so it is linear.
• If a = 0, the x² term vanishes and it is no longer quadratic — correct. Yes! Then only bx + c is left.
• Because a must be positive. a can be negative, like −2x² + 1. Only a = 0 is not allowed.
• Because b and c cannot be 0. b or c may be 0. y² − 2 is quadratic with b = 0. Only a must not be 0.
• 4x + 2. A polynomial of degree 1.
• 2y² − 3y + 4. A polynomial of degree 2.
• 1/(x − 1) — correct. Yes! The book lists 1/(x − 1) and √x + 2 as examples that are not polynomials.
• 7u⁶ − (3/2)u⁴ + 4u² + u − 8. A polynomial of degree 6.