Class 10 Maths Notes Chapter 2 — Polynomials#
Key terms#
- Polynomial in $x$ — an expression $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$, where the powers are whole numbers and the coefficients are real numbers.
- Degree — the highest power of $x$ in $p(x)$.
- Zero of $p(x)$ — a real number $k$ such that $p(k) = 0$.
| Type | Degree | General form |
|---|---|---|
| Linear | 1 | $ax + b$, $a \neq 0$ |
| Quadratic | 2 | $ax^2 + bx + c$, $a \neq 0$ |
| Cubic | 3 | $ax^3 + bx^2 + cx + d$, $a \neq 0$ |
- The zero of the linear polynomial $ax + b$ is $-\frac{b}{a}$.
- A polynomial of degree $n$ has at most $n$ zeroes. A quadratic has at most 2, a cubic at most 3.
Geometrical meaning of zeroes#
The zeroes of $p(x)$ are exactly the x-coordinates of the points where the graph of $y = p(x)$ meets the x-axis.
- Linear ($y = ax + b$): a straight line that cuts the x-axis at exactly one point, $\left(-\frac{b}{a}, 0\right)$. So it has one zero.
- Quadratic ($y = ax^2 + bx + c$): a curve called a parabola. It opens upward if $a > 0$ and downward if $a < 0$.
| Graph of a quadratic | Number of zeroes |
|---|---|
| Cuts the x-axis at two points | 2 (distinct) |
| Touches the x-axis at one point | 1 (two equal zeroes) |
| Does not meet the x-axis | 0 |
- Cubic: the graph meets the x-axis at 1, 2 or 3 points, so a cubic has 1, 2 or 3 zeroes.
- To count zeroes from a graph, count the points where it meets the x-axis.
(Plotting graphs of quadratic or cubic polynomials is not required of students.)
Relationship between zeroes and coefficients#
Quadratic $ax^2 + bx + c$ with zeroes $\alpha$, $\beta$:
$$\alpha + \beta = -\frac{b}{a}$$$$\alpha\beta = \frac{c}{a}$$
Cubic $ax^3 + bx^2 + cx + d$ with zeroes $\alpha$, $\beta$, $\gamma$:
$$\alpha + \beta + \gamma = -\frac{b}{a}$$$$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$$
$$\alpha\beta\gamma = -\frac{d}{a}$$
In words, for a quadratic:
Sum of zeroes = $-\dfrac{\text{coefficient of } x}{\text{coefficient of } x^2}$
Product of zeroes = $\dfrac{\text{constant term}}{\text{coefficient of } x^2}$
Forming a quadratic from the sum $S$ and product $P$ of its zeroes:
$$p(x) = k(x^2 - Sx + P)$$where $k$ is any non-zero real number. Usually we take $k$ to clear fractions.
Example: zeroes of $x^2 - 2x - 8$.
$x^2 - 2x - 8 = (x - 4)(x + 2)$, so the zeroes are 4 and −2.
Check: $\alpha + \beta = 4 + (-2) = 2 = -\frac{-2}{1}$ and $\alpha\beta = 4 \times (-2) = -8 = \frac{-8}{1}$.
Common mistakes#
- Forgetting the minus sign: $\alpha + \beta = -\frac{b}{a}$, not $\frac{b}{a}$.
- In a cubic, the product is $\alpha\beta\gamma = -\frac{d}{a}$ (with a minus), unlike the quadratic $\alpha\beta = \frac{c}{a}$.
- Reading coefficients from a polynomial not in standard form. Write $6x^2 - 3 - 7x$ as $6x^2 - 7x - 3$ first.
- Missing a zero coefficient: for $t^2 - 15$, $b = 0$, so $\alpha + \beta = 0$.
- A graph that only touches the x-axis still gives a zero.