What you will learn
- Recognise a polynomial
- Identify terms, coefficients and degree
- Evaluate a polynomial
- Distinguish linear, quadratic and cubic polynomials
What is a polynomial?
A polynomial is an algebraic expression made from variables, constants and whole-number powers of variables. Its terms are joined using addition or subtraction.
Expressions such as 3x² − 5x + 2 and 7 are polynomials. An expression containing 1/x is not a polynomial in x because it uses the negative power x⁻¹.
Is 4y³ − 2y + 9 a polynomial?
Solution: Yes. Every exponent of y is a non-negative whole number.
Terms, coefficients and degree
In 5x² − 3x + 8, the terms are 5x², −3x and 8. The coefficients of x² and x are 5 and −3, while 8 is the constant term.
The degree is the highest exponent with a non-zero coefficient. The degree of 5x² − 3x + 8 is 2, so it is quadratic.
Find the degree of 2a⁴ + a² − 6.
Solution: The highest exponent is 4, so the degree is 4.
Finding a polynomial's value
To evaluate a polynomial, replace its variable with the given number and simplify carefully. Use brackets when substituting a negative value.
For p(x) = x² + 2x − 3, p(2) means 2² + 2×2 − 3 = 5.
For p(x) = x² − 4, find p(−3).
Solution: p(−3) = (−3)² − 4 = 9 − 4 = 5.
Key terms
- Term
- One part of an expression separated by plus or minus signs.
- Coefficient
- The numerical factor multiplying a variable term.
- Degree
- The greatest exponent with a non-zero coefficient.
Common mistakes
- Treating a negative exponent as allowed in a polynomial.
- Forgetting that a non-zero constant polynomial has degree 0.
- Dropping brackets while substituting a negative value.
Questions and answers
1. What is the degree of 7x³ − x + 4?
Answer: 3
2. Is 2/x + 1 a polynomial in x?
Answer: No, because 2/x = 2x⁻¹ has a negative exponent.
3. If p(x) = 2x² − 3x + 1, find p(−1).
Answer: 6
2(−1)² − 3(−1) + 1 = 2 + 3 + 1 = 6.
One-minute revision
- Polynomial exponents are non-negative whole numbers.
- Highest exponent gives the degree.
- Substitute with brackets, then follow operation order.