Polynomials: degree, types and zeroes
A polynomial's degree is its highest power of x. A zero of p(x) is a value k where p(k) = 0.
What a polynomial is
A polynomial in x is a sum of terms like 3x², −5x and 7. In every term, the power of x must be a whole number (0, 1, 2, 3, ...). So 2x³ − x + 4 is a polynomial. But x + 1/x and √x + 2 are not, because 1/x = x⁻¹ and √x has power 1/2. The coefficient of a term is the number that multiplies the power of x. The degree is the highest power of x with a non-zero coefficient.
Types of polynomials by degree
| Degree | Name | General form | Example |
|---|---|---|---|
| 0 | Constant | c (c ≠ 0) | 7 |
| 1 | Linear | ax + b, a ≠ 0 | 2x − 3 |
| 2 | Quadratic | ax² + bx + c, a ≠ 0 | x² − 5x + 6 |
| 3 | Cubic | ax³ + bx² + cx + d, a ≠ 0 | x³ − 4x |
Example 1: value and zeroes of a polynomial
For p(x) = x² + x − 20, find p(2). Check whether 4 and −5 are zeroes of p(x).
- Replace x by 2: p(2) = 2² + 2 − 20 = 4 + 2 − 20 = −14
- p(4) = 4² + 4 − 20 = 16 + 4 − 20 = 0, so 4 is a zero.
- p(−5) = (−5)² + (−5) − 20 = 25 − 5 − 20 = 0, so −5 is a zero.
p(2) = −14. Both 4 and −5 are zeroes of p(x).
Example 2: zero of a linear polynomial
Find the zero of p(x) = 3x + 12.
- Rule: the zero of ax + b is −b/a.
- Here a = 3 and b = 12.
- Zero = −12/3 = −4
- Check: p(−4) = 3(−4) + 12 = −12 + 12 = 0.
The zero of 3x + 12 is −4.
Quick facts
- A zero of p(x) is a number k such that p(k) = 0.
- A linear polynomial ax + b (a ≠ 0) has exactly one zero, −b/a.
- A polynomial of degree n has at most n zeroes: a quadratic at most 2, a cubic at most 3.
- A non-zero constant polynomial, like 7, has no zero.
- By number of terms: a monomial has one term, a binomial two and a trinomial three.
A tea stall's profit, in rupees, for selling x cups might be 8x − 200. Here ₹ 200 is an example daily cost. Its zero, x = 25, is the number of cups needed just to break even.
1 marks: “What is the degree of the polynomial 5 − 2x² + 7x³ − x?” The highest power of x is 3, so the degree is 3.
2 marks: “If x = 2 is a zero of p(x) = x² + kx − 6, find k.” p(2) = 4 + 2k − 6 = 0, so 2k = 2 and k = 1. Check: x² + x − 6 = (x + 3)(x − 2).
1 marks: “Which of these is not a polynomial: x² − 1, 3x + √2, x + 1/x or 7?” x + 1/x, because 1/x = x⁻¹ and −1 is not a whole number.
Degree = the top power. Zero = the x that makes the whole thing zero.