x class polynomials
Polynomials
What is polnyomial?
Polynomial is an algebraic expression constructed using constants and variables.Coefficients operate on variables, which can be raised to various powers of non-negative integer exponents.
Example: 2x + 5, 3x2 + 5x + 6, - 5y, x3 are some polynomials.
`1/x^2,1/sqrt (2x),1/(y-1),sqrt (3x^3)` are not polynomials.
General form of a polynomial :
An algebraic expression of the form
P(x) =a0xn+ a1xn-1+ a2xn-2+ a3xn-3+ - - - - - - + an-1x+ an where a0, a1, a2,- - - an-1, an are real numbers, n is a non-negative integer and a0 ≠ 0 is called a polynomial of degree n.
Degree of polynomial:
The highest power of x in a polynomial p (x) is called the degree of polynomial.
Example:p(u)=7u 6–3u 4 + 4u 2 – 8 is a polynomial of degree 6
p(x)=x10 – 3x8 + 4x5 + 2x2 -1 is a polynomial of degree 10.
Types of polynomial:
Constant polynomial:
A polynomial of degree zero is called a constant polynomial or zero degree polynomial and it is in the form of p (x) = k.
Linear polynomial (First degree polynomial):
A polynomial of degree one is called linear polynomial and it is of the form p (x) = ax + b, where a, b are real numbers and a ≠ 0.
Quadratic polynomial:
A polynomial of degree two is called quadratic polynomial and it is of the form p (x) = ax2 + bx + c, where a, b, c are real numbers and a ≠ 0
Example: x2+ 5x + 4
Cubic polynomial:
A polynomial of degree three is called cubic polynomial and it is of the form p (x) = ax3 + bx2 + cx + d where a, b, c, d are real numbers and a ≠ 0.
Example: 5x3–4x2+x–1
Value of a polynomial:
If p(x) is a polynomial in x, and if k is a 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: p(x) = x2 – 2x – 3, what is the value at x = 1?
Putting x = 1, in the polynomial, we get
p(1) = (1)2 – 2(1) – 3 = 1–2–3 = –4.
The value = – 4
This is the value of p(x) at x = 1
Zeroes of a polynomial:
A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.
Example: What are the values of p(x) = x2 – 2x – 3 at x = 3, -1 and 2?
p(3) = (3)2 – 2(3) – 3 = 9 – 6 – 3 = 0
p(-1) = (–1)2 – 2(–1) – 3 = 1 + 2 – 3 = 0
p(2) = (2)2 – 2(2) – 3 = 4 – 4 – 3 = –3
We see that p(3) = 0 and p(-1) = 0. These points, x = 3 and x = –1, are called Zeroes of the polynomial p(x) = x2 – 2x -3.
Exercise-1 >
As p(2) ≠ 0, 2 is not the zero of p(x).
If k is a zero of p(x) = ax+b, a ≠ 0.
then p(k) = ak + b = 0,
i.e., k = `(-b)/a`
The zero of the linear polynomial ax + b is `(-b)/a`.
Graph of 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)
Example:Draw the graph of the polynomial f(x) = 2x +3. Also, find the coordinates of the point where it crosses X-axis.
Solution:Let y = 2x +3.
The following table list the values of y corresponding to different values of x.
| x | -2 | 0 | 2 |
|---|---|---|---|
| y = 2x +3 | -1 | 3 | 7 |
| (x, y) | (–2, –1) | (0, 3) | (2, 7) |
The points (–2, –1), (0, 3) and (2, 7) are plotted on the graph paper on a suitable scale. A line is drawn passing through these points to obtain the graphs of the given polynomial.
From the graph, you can see that the graph of y = 2x+3 intersects the x-axis between x = –1 and x = –2, that is, at the point (`-3/2`,0 )
☘ Graph of a quadratic polynomial p (x) = ax2+ bx + c is a parabola which open upwards like ∪ if a > 0.
Example:Draw the graph of the polynomial f(x) = x2 - 3x - 4
Solution:The following table gives the values of y or f(x) for various values of x.
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|
| y = x2 – 3x – 4 | 6 | 0 | -4 | -6 | -6 | -4 | 0 | 6 |
| (x, y) | (– 2, 6) | (– 1, 0) | (0, 4) | (1, – 6) | (2, – 6) | (3, – 4) | (4, 0) | (5, 6) |
Let us plot the points (– 2, 6) ,(– 1, 0), (0, 4), (1, – 6), (2, – 6), (3, – 4), (4, 0) and (5, 6) on a graphs paper and draw a smooth free hand curve passing through these points.
The curve thus obtained represents the graphs of the polynomial f(x) = x2 - 3x - 4.It is like a shaped ∪ curve. This is called a parabola.
☘ Graph of a quadratic polynomial p(x) = ax2 + bx + c is a parabola which open downwards like ∩ if a< 0.
☘ These curves are called parabolas.
☘ The zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x-axis.
In general, a polynomial p (x) of degree n crosses the x -axis at, at most n points.
Geometrically, the zeroes of a polynomial p(x) are the x coordinates of the points, where the graph of y = p(x) intersects the x-axis.
A polynomial of degree ‘n’ can have at most n zeroes.
That is, a quadratic polynomial can have at most 2 zeroes
A cubic polynomial can have at most 3 zeroes.
Discriminant of a quadratic polynomial:
For polynomial p(x) = ax2 + bx + c, a ≠ 0, the expression b2 – 4ac is known as its discriminant ‘ D’ or Δ.
D = b2– 4ac or Δ= b2 – 4ac
֍ If D > 0, graph of p(x) = ax2 + bx + c will intersect the x -axis at two distinct points.
The x coordinates of points of intersection with x-axis are known as ‘zeroes’ of p (x).
֍ If D = 0, graph of p(x) = ax2 + bx + c will touch the x-axis at exactly one point. p (x) will have only one ‘zero’.
֍ If D< 0, graph of p(x) = ax2 + bx + c will neither touch nor intersect the x-axis. p (x) will not have any real ‘zero’.
| Nature of the roots | Figure if a < 0 | Figure if a > 0 |
Δ= b2 – 4ac > 0 roots are real and un-equal number of zeroes is 2 |
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Δ= b2 – 4ac = 0 roots are real and equal number of zeroes is 1 |
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Δ= b2 – 4ac < 0 roots are imaginary or complex number of zeroes is 0 |
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Exercise-2 >
Relationship between the zeroes and the coefficients of a polynomial:
If α, β are zeroes of p (x) = ax2 + bx + c, then
Sum of zeroes = α + β = `-b/a = -(coefficient of x)/(coefficient of x^2)`
Product of zeroes = α β = `c/a = (constant term)/(coefficient of x^2)`
If α, β, γ are zeroes of p (x) = ax3 + bx2+ cx + d, then
α + β + γ = `-b/a = -(coefficient of x^2)/(coefficient of x^3 )`
α β + β γ + γ α = `c/a = (coefficient of x)/(coefficient of x^3)`
α β γ = `-d/a=-(constant term)/(coefficient of x^3)`
֍ If α, β are roots of a quadratic polynomial p (x), then
p (x) = x2 – (sum of zeroes) x + product of zeroes
p (x) = x2 – (α + β) x + α β
֍ If α, β, γ are the roots of a cubic polynomial p (x), then
p (x) = x3 – (sum of zeroes) x2 + (sum of product of zeroes taken two at a time) x – product of zeroes
p (x) = x3 – (α + β + γ) x2 + (α β + β γ + γ α) x – α β γ
Exercise-3 >
Division algorithm for polynomials:
If p (x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q (x) and r (x) such that p (x) = q (x) . g (x) + r (x), where r (x) = 0 or degree of r (x) < degree of g (x).
(or) Dividend = Quotient `times` Divisor + Remainder
Here p(x) is the dividend
g (x) is the divisor
q(x) is the quotient and
r (x) is the remainder.
Exercise-4 >
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