Algebra
Polynomials
Polynomial basics
20 practice questions
2 video lessons
Theory + worked examples
Theory
A polynomial is a sum of terms \(ax^n\) with whole-number exponents:
- Degree: the highest power.
- Leading coefficient: the coefficient of the highest-power term.
- Standard form: terms in descending order of degree.
Name by degree (linear, quadratic, cubic) and by terms (monomial, binomial, trinomial).
The parts of a polynomial.
Naming polynomials.
General form:
\[a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\]
The degree sets the end behavior and max number of zeros.
How to analyze a polynomial
- Find the highest power for the degree.
- Read the leading coefficient.
- Write in descending order (standard form).
- Classify by degree and term count.
Example 1 — Degree
What is the degree of \(3x^4+2x-1\)?
Solution
The highest power.
| \(\text{degree}\) | \(=\) | \(4\) |
Example 2 — Leading coefficient
What is the leading coefficient of \(5x^3-2x^2+1\)?
Solution
The coefficient of the highest-power term.
| \(\text{leading coefficient}\) | \(=\) | \(5\) |
Example 3 — Standard form
Write \(2x-5+x^3\) in standard form.
Solution
Descending powers.
| \(x^3+2x-5\) |
Example 4 — Classify
Classify \(4x^2+3\) by degree and number of terms.
Solution
Degree \(2\), two terms.
| \(\text{quadratic binomial}\) |
Common pitfalls
Degree is the highest power, not the number of terms.
Exponents must be whole numbers for a polynomial.
Standard form is descending, highest power first.
Frequently asked questions
What is the degree of a polynomial?
The highest power of the variable.
What is the leading coefficient?
The coefficient of the highest-power term.
What is standard form?
Terms written in descending order of degree.
What is a binomial?
A polynomial with two terms.
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