Polynomials
Relationship between zeroes and coefficients
Video Lectures
● Part -1 (Geometrical Meaning of the Zeroes of a Polynomials )
● Part -1 (Geometrical Meaning of the Zeroes of a Polynomials )
Polynomial Let `x` be a variable, `n` be a whole number, and `a_1, a_2, a_3 ,....., a_(n-1), a_n` be constants (real numbers). Then `f(x) = a_nx^n + a_(n-1)x^(n-1) + ......+ a_1x + a_0` is called a polynomial in variable `x`.
In the polynomial `f(x) = a_nx^n + a_(n-1)x^(n-1) + ......+ a_1x + a_0`, `a_nx^n , a_(n-1)x^(n-1), ...... a_1x , and a_0` are known as the terms of the polynomials and `a_n, a_(n-1), ......., a_1, and a_0` are their coefficients.
Examples
`f(x) = 3x -2` is a polynomial in variable `x`.
`p(y) = 3y^2 -2y + 4` is a polynomial in variable `y`.
Degree of the polynomial If `p(x)` is a polynomial in `x`, the highest power of `x` in `p(x)` is called the degree of the polynomial `p(x)`.
Examples
`4x + 2` is a polynomial in the variable `x` of degree 1.
`2y^2 – 3y + 4` is a polynomial in the variable `y` of degree 2.
Linear Polynomial A polynomial of degree 1 is called a linear polynomial. (General form `f(x) = ax +b`) For example, `2x – 3, \sqrt 2x +14`, etc.
Quadratic polynomial A polynomial of degree 2 is called a quadratic polynomial. (General form `f(x) = ax^2 +bx + c`) The name ‘quadratic’ has been derived from the word ‘quadrate’, which means ‘square’. For example, `2x^2 + 5, y^2 -2` , etc.
Cubic polynomial A polynomial of degree 3 is called a cubic polynomial. (General form `f(x) = ax^3 +bx^2 + cx + d`). Some examples `x^3 -2, 2x^3 + 4x^2 + 2x + 1`, etc.
Value of the polynomial If `p(x`) is a polynomial in `x`, and if `k` is any real number, then the value obtained by replacing `x` by `k` in `p(x)`, is called the value of `p(x)` at `x = k`, and is denoted by `p(k)`.
Example
What is the value of `p(x) = x^2 –3x – 4` at `x = 2`?
`p(2) = 2^2 -3 × 2 -4`
`p(2) = 4 -6 -4`
`p(2) = 2`
Geometrical Meaning of the Zeroes of a Polynomial
The zeroes of a polynomial `p(x)` are precisely the `x`-coordinates of the points, where the
graph of `y = p(x)` intersects the `x` -axis
1. Linear Polynomial
In general, for a linear polynomial `ax + b, a ≠ 0`, the graph of `y = ax + b` is a
straight line which intersects the `x`-axis at exactly one point, namely `(-b/a, 0)`.
Therefore, the linear polynomial `ax + b, a ≠ 0`, has exactly one zero, namely, the `x`-coordinate of the point where the graph of `y = ax + b` intersects the `x`-axis.
Example:
The zero of `2x + 3` is `-3/2` . Thus, the zero of
the polynomial `2x + 3` is the `x`-coordinate of the point where the
graph of `y = 2x + 3` intersects the `x`-axis.
2. Quadratic polynomial
For any quadratic
polynomial `ax^2
+ bx + c, a ≠ 0`, the
graph of the corresponding
equation `y = ax^2
+ bx + c` has one
of the two shapes either open
upwards or open
downwards depending on
whether `a > 0 or a < 0`. (These
curves are called parabolas.)
Example:
Three cases
Case (i): Here, the graph cuts `x`-axis at two distinct points `A` and `A′`.
The `x`-coordinates of `A` and `A′` are the two zeroes of the quadratic polynomial `ax^2
+ bx + c` in this case
➤ If `alpha` and `beta` are the zeroes of the quadratic polynomial `ax^2
+ bx + c`, then,
`alpha + beta = -b/a, alpha beta = c/a `