Class 10 Maths · Chapter 2 · Topic 1 of 4
What makes an expression a polynomial?
Simplify an expression before naming its degree.
Ideas you’ll use
- Polynomial
- An expression made by adding terms such as 3x², −2x and 5. Powers of x must be whole numbers: 0, 1, 2, …; coefficients can be any real numbers.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
Different degrees make different shapes
p(x) (number)
x (number)
- x · Original teaching example; exact values are available below.
- x² · Original teaching example; exact values are available below.
- x³ · Original teaching example; exact values are available below.
Assumptions: Lines join calculated sample points. Curves are visual approximations; use the formula and marked exact values for proof.
Read the plotted values
x
| x (number) | p(x) (number) |
|---|---|
| -2 | -2 |
| -1.9 | -1.9 |
| -1.8 | -1.8 |
| -1.7 | -1.7 |
| -1.6 | -1.6 |
| -1.5 | -1.5 |
| -1.4 | -1.4 |
| -1.3 | -1.3 |
| -1.2 | -1.2 |
| -1.1 | -1.1 |
| -1 | -1 |
| -0.9 | -0.9 |
| -0.8 | -0.8 |
| -0.7 | -0.7 |
| -0.6 | -0.6 |
| -0.5 | -0.5 |
| -0.4 | -0.4 |
| -0.3 | -0.3 |
| -0.2 | -0.2 |
| -0.1 | -0.1 |
| 0 | 0 |
| 0.1 | 0.1 |
| 0.2 | 0.2 |
| 0.3 | 0.3 |
| 0.4 | 0.4 |
| 0.5 | 0.5 |
| 0.6 | 0.6 |
| 0.7 | 0.7 |
| 0.8 | 0.8 |
| 0.9 | 0.9 |
| 1 | 1 |
| 1.1 | 1.1 |
| 1.2 | 1.2 |
| 1.3 | 1.3 |
| 1.4 | 1.4 |
| 1.5 | 1.5 |
| 1.6 | 1.6 |
| 1.7 | 1.7 |
| 1.8 | 1.8 |
| 1.9 | 1.9 |
| 2 | 2 |
x²
| x (number) | p(x) (number) |
|---|---|
| -2 | 4 |
| -1.9 | 3.61 |
| -1.8 | 3.24 |
| -1.7 | 2.89 |
| -1.6 | 2.56 |
| -1.5 | 2.25 |
| -1.4 | 1.96 |
| -1.3 | 1.69 |
| -1.2 | 1.44 |
| -1.1 | 1.21 |
| -1 | 1 |
| -0.9 | 0.81 |
| -0.8 | 0.64 |
| -0.7 | 0.49 |
| -0.6 | 0.36 |
| -0.5 | 0.25 |
| -0.4 | 0.16 |
| -0.3 | 0.09 |
| -0.2 | 0.04 |
| -0.1 | 0.01 |
| 0 | 0 |
| 0.1 | 0.01 |
| 0.2 | 0.04 |
| 0.3 | 0.09 |
| 0.4 | 0.16 |
| 0.5 | 0.25 |
| 0.6 | 0.36 |
| 0.7 | 0.49 |
| 0.8 | 0.64 |
| 0.9 | 0.81 |
| 1 | 1 |
| 1.1 | 1.21 |
| 1.2 | 1.44 |
| 1.3 | 1.69 |
| 1.4 | 1.96 |
| 1.5 | 2.25 |
| 1.6 | 2.56 |
| 1.7 | 2.89 |
| 1.8 | 3.24 |
| 1.9 | 3.61 |
| 2 | 4 |
x³
| x (number) | p(x) (number) |
|---|---|
| -2 | -8 |
| -1.9 | -6.859 |
| -1.8 | -5.832 |
| -1.7 | -4.913 |
| -1.6 | -4.096 |
| -1.5 | -3.375 |
| -1.4 | -2.744 |
| -1.3 | -2.197 |
| -1.2 | -1.728 |
| -1.1 | -1.331 |
| -1 | -1 |
| -0.9 | -0.729 |
| -0.8 | -0.512 |
| -0.7 | -0.343 |
| -0.6 | -0.216 |
| -0.5 | -0.125 |
| -0.4 | -0.064 |
| -0.3 | -0.027 |
| -0.2 | -0.008 |
| -0.1 | -0.001 |
| 0 | 0 |
| 0.1 | 0.001 |
| 0.2 | 0.008 |
| 0.3 | 0.027 |
| 0.4 | 0.064 |
| 0.5 | 0.125 |
| 0.6 | 0.216 |
| 0.7 | 0.343 |
| 0.8 | 0.512 |
| 0.9 | 0.729 |
| 1 | 1 |
| 1.1 | 1.331 |
| 1.2 | 1.728 |
| 1.3 | 2.197 |
| 1.4 | 2.744 |
| 1.5 | 3.375 |
| 1.6 | 4.096 |
| 1.7 | 4.913 |
| 1.8 | 5.832 |
| 1.9 | 6.859 |
| 2 | 8 |
These are three non-zero polynomials of degrees 1, 2 and 3. The picture supports the algebraic definition; shape alone does not define degree.
Check the powers
A polynomial uses non-negative whole-number powers of x. Terms like 1/x or √x do not fit.
x², x and 1 are allowedIn the picture: Compare the linear, quadratic and cubic curves.
Read all the reasoning together
Check the powers
A polynomial uses non-negative whole-number powers of x. Terms like 1/x or √x do not fit.
x², x and 1 are allowedCombine like terms
Collect terms with the same power. A cancelled term no longer contributes to the degree.
3x²+2x−3x²+7 = 2x+7Read the highest remaining power
The highest power with a non-zero coefficient is the degree. A non-zero constant has degree zero.
degree(2x+7)=1; degree(7)=0
Simplify an expression before naming its degree.
Worked example: Simplify before naming the degree
3x² + 2x − 3x² + 7 = 2x + 7The x² terms cancel, so the degree is 1, not 2. The non-zero constant polynomial 7 has degree 0.
Read the supporting explanation
A coefficient is the number multiplying a power of x. The degree is the highest power left after like terms are combined. Degree 1 is linear, degree 2 is quadratic and degree 3 is cubic.
Go deeper: What about the zero polynomial?
The zero polynomial has no non-zero term, so its degree is not defined. It is zero at every x. The usual bound on the number of zeroes applies to non-zero polynomials.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Is (√2)x² + 5 a polynomial? What about √x + 5?
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