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Linear Polynomial — Definition, Formula & Graph

Linear Polynomial

A polynomial with degree 1. For example, the following are all linear polynomials: 3x + 5, y – ½, and a.

 

 

See also

Linear

Key Formula

p(x)=ax+bp(x) = ax + b
Where:
  • aa = The leading coefficient (must not be zero)
  • bb = The constant term (can be any real number, including zero)
  • xx = The variable

Worked Example

Problem: Determine whether p(x) = 4x + 7 is a linear polynomial, and find its root.
Step 1: Identify the highest power of x in the polynomial.
p(x)=4x1+7p(x) = 4x^1 + 7
Step 2: The highest exponent is 1, so the degree is 1. This confirms it is a linear polynomial.
Step 3: To find the root, set p(x) = 0 and solve for x.
4x+7=04x + 7 = 0
Step 4: Subtract 7 from both sides.
4x=−74x = -7
Step 5: Divide both sides by 4.
x=−74x = -\frac{7}{4}
Answer: p(x) = 4x + 7 is a linear polynomial (degree 1), and its root is x = −7/4.

Another Example

Problem: Is q(x) = 6x² − x + 2 a linear polynomial?
Step 1: Identify the highest power of x in the polynomial.
q(x)=6x2−x+2q(x) = 6x^2 - x + 2
Step 2: The highest exponent is 2, so the degree is 2. A degree-2 polynomial is called a quadratic polynomial, not a linear polynomial.
Answer: No. q(x) = 6x² − x + 2 has degree 2, so it is a quadratic polynomial, not a linear polynomial.

Frequently Asked Questions

Is a constant like 5 a linear polynomial?
No. A constant by itself, such as 5, is a polynomial of degree 0 (called a constant polynomial). A linear polynomial must have a variable term with exponent 1, like 5x or 5x + 3.
How many roots does a linear polynomial have?
A linear polynomial has exactly one root. Because the general form is ax + b with a ≠ 0, you can always solve for a single value: x = −b/a. This also means its graph crosses the x-axis at exactly one point.

Linear polynomial vs. Quadratic polynomial

A linear polynomial has degree 1 and its graph is a straight line. It always has exactly one root. A quadratic polynomial has degree 2 and its graph is a parabola. A quadratic can have zero, one, or two real roots. For example, 3x + 2 is linear while 3x² + 2 is quadratic.

Why It Matters

Linear polynomials are the simplest polynomials that contain a variable, making them the foundation for solving basic equations and understanding functions. Every linear equation you encounter in algebra—from calculating speed to converting temperatures—relies on a linear polynomial. They also appear when you factor higher-degree polynomials into simpler pieces.

Common Mistakes

Mistake: Calling a constant like 7 a linear polynomial because it looks simple.
Correction: A constant has degree 0, not degree 1. A linear polynomial must include a variable raised to the first power, such as 7x or 7x + 3.
Mistake: Thinking that any polynomial containing an x term is linear, even if higher powers of x are present.
Correction: The degree of a polynomial is determined by its highest-power term. For example, x² + x has degree 2 (quadratic), not degree 1, even though it contains an x term.

Related Terms