Chapter 2: Polynomials
1. Key Ideas
A polynomial is an algebraic expression in which the powers of the variable are non-negative integers.
The degree of a polynomial is the highest power of the variable with a non-zero coefficient.
Types by degree
- Linear polynomial: degree 1
General form: (ax+b,\ a\neq0) - Quadratic polynomial: degree 2
General form: (ax^2+bx+c,\ a\neq0) - Cubic polynomial: degree 3
General form: (ax^3+bx^2+cx+d,\ a\neq0)
2. Value and Zero of a Polynomial
If (p(x)) is a polynomial, then its value at (x=k) is obtained by substituting (k) for (x). It is written as (p(k)).
A real number (k) is called a zero of (p(x)) if
[
p(k)=0.
]
For example, for
[
p(x)=x^2-3x-4,
]
we get
[
p(-1)=0,\qquad p(4)=0.
]
Therefore, (-1) and (4) are its zeroes.
Important point
For a linear polynomial (ax+b), its zero is
[
\boxed{-\frac{b}{a}}.
]
3. Geometrical Meaning of Zeroes
The graph of
[
y=p(x)
]
represents the polynomial geometrically.
The zeroes of a polynomial are exactly the x-coordinates of the points where its graph intersects the x-axis.
Linear polynomial
The graph of (y=ax+b) is a straight line and intersects the x-axis at exactly one point.
Hence, a linear polynomial has exactly one zero.
Quadratic polynomial
The graph of
[
y=ax^2+bx+c
]
is a parabola.
- If (a>0), it opens upwards.
- If (a<0), it opens downwards.
A quadratic polynomial can have:
- Two distinct zeroes — graph cuts the x-axis at two points.
- Two equal zeroes — graph touches the x-axis at one point.
- No real zero — graph does not meet the x-axis.
Therefore, a quadratic polynomial has at most two zeroes.
Cubic polynomial
A cubic polynomial can have at most three zeroes.
In general:
[
\boxed{\text{A polynomial of degree }n\text{ has at most }n\text{ zeroes.}}
]
4. Relationship Between Zeroes and Coefficients
This is one of the most important results of the chapter.
Quadratic Polynomial
Let
[
p(x)=ax^2+bx+c,\qquad a\neq0
]
and let its zeroes be (\alpha) and (\beta).
Then
[
\boxed{\alpha+\beta=-\frac{b}{a}}
]
and
[
\boxed{\alpha\beta=\frac{c}{a}}.
]
Easy way to remember
| Quantity | Formula |
|---|---|
| Sum of zeroes | (-\frac{\text{coefficient of }x}{\text{coefficient of }x^2}) |
| Product of zeroes | (\frac{\text{constant term}}{\text{coefficient of }x^2}) |
These formulas allow the relationship between the zeroes and coefficients to be verified without finding the zeroes again.
5. Forming a Quadratic Polynomial from Its Zeroes
Suppose the zeroes are (\alpha) and (\beta).
A quadratic polynomial having these zeroes can be written as
[
\boxed{k(x-\alpha)(x-\beta)},\qquad k\neq0.
]
If only the sum (S) and product (P) of the zeroes are known, one convenient polynomial is
[
\boxed{x^2-Sx+P}.
]
For example, if
[
\alpha+\beta=-3,\qquad \alpha\beta=2,
]
then
[
x^2-(-3)x+2=x^2+3x+2.
]
Any non-zero constant multiple of this polynomial has the same zeroes.
6. Cubic Polynomial
Consider
[
p(x)=ax^3+bx^2+cx+d,\qquad a\neq0.
]
If its zeroes are (\alpha,\beta,\gamma), then:
Sum of zeroes
[
\boxed{\alpha+\beta+\gamma=-\frac{b}{a}}
]
Sum of products taken two at a time
[
\boxed{\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}}
]
Product of zeroes
[
\boxed{\alpha\beta\gamma=-\frac{d}{a}}
]
Sign pattern to remember
For
[
ax^3+bx^2+cx+d,
]
the relationships are
[
\boxed{
\alpha+\beta+\gamma=-\frac ba,\quad
\alpha\beta+\beta\gamma+\gamma\alpha=\frac ca,\quad
\alpha\beta\gamma=-\frac da
}
]
The signs alternate in these coefficient relationships.
7. Factorisation and Zeroes
If a polynomial can be factorised, its zeroes can often be found directly.
For example,
[
2x^2-8x+6
=2(x-1)(x-3).
]
Therefore,
[
x=1,\quad x=3
]
are the zeroes.
The basic idea is:
[
\boxed{\text{If }p(x)=(x-\alpha)(x-\beta)\times\text{constant, then }\alpha,\beta\text{ are zeroes.}}
]
8. Quick Revision Sheet
Definitions
- Degree = highest power of the variable.
- Degree 1 → linear
- Degree 2 → quadratic
- Degree 3 → cubic
- Zero (k) of (p(x)) means (p(k)=0).
Graph connection
[
\boxed{\text{Zeroes = x-coordinates where the graph meets the x-axis}}
]
Maximum number of zeroes
- Linear → at most 1
- Quadratic → at most 2
- Cubic → at most 3
- Degree (n) → at most (n)
Quadratic (ax^2+bx+c)
If zeroes are (\alpha,\beta):
[
\boxed{\alpha+\beta=-\frac ba}
]
[
\boxed{\alpha\beta=\frac ca}
]
Cubic (ax^3+bx^2+cx+d)
If zeroes are (\alpha,\beta,\gamma):
[
\boxed{\alpha+\beta+\gamma=-\frac ba}
]
[
\boxed{\alpha\beta+\beta\gamma+\gamma\alpha=\frac ca}
]
[
\boxed{\alpha\beta\gamma=-\frac da}
]