Chapter 2 : Polynomial
What is polynomial?
A polynomial is an expression made up of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.
Example : 3x4-2x2+7x-5.
Key Features of Polynomial :
let's discuss some features of polynomial:
Take a polynomial : 3x4-2x2+7x-5.
a) Degree: The highest power of the variable(x)(e.g. degree of 3x4-2x2+7x-5 is 4).
b) Terms: Each part separated by + or – (e.g., 3x4, -2x2, 7x, -5).
c) Coefficients: Numbers multiplying the variables (e.g., 3, -2, 7).
d) Constant term: The standalone number (e.g. -5).
Types of polynomial:
a) Linear polynomial: A polynomial of degree 1.
Example : 2x – 3 , u + 5 , 3u+ 5
b) Quadratic polynomial : A polynomial of degree 2
Example : 3x2+2x – 3 ,3u2 - u + 5
c) Cubic polynomial : A polynomial of degree 3
Example : 2x3+ 3x2+2x – 3 , 4u3 - 3u2 - u + 5
General form of polynomial:
Quadratic polynomial : ax2 + bx + c where a, b c are real numbers and a ≠ 0.
Cubic polynomial : ax3 + bx2 + cx + d where a, b c ,d are real numbers and a ≠ 0.
Zeros of polynomial:
Suppose P(x) is a quadratic polynomial , so P(x) = ax2 + bx + c , now there is a value of x such that P(x) = 0 , that value of x is called zero of polynomial.
NOTE: Number of zeros depends on the degree of polynomial ( Number of zeroes = degree of polynomial)
Geometrical Meaning of the Zeroes of a Polynomial
If we plot the polynomial P(x) against y on x- y plane , and the curve cuts the x- axis , then the values of x at which it cut the x- axis are zeroes of that polynomial.
Example : P(x) = x2+ 2x -4
In the above graph , curve cut the x-axis on two points. It means there are two zeros of given polynomial.
Example 2 : P(x) = x2+ 2x +2
In this example, curve does not cut the x-axis at all , it means there are no real zeros possible for this polynomial.
Example 3: P(x) = x2 + 2x +1
In the above graph, curve cuts the graph at one point , it means it has two equal roots (quadratic equal , that's way it has two zeros)
Relationship between Zeroes and Coefficients of a Polynomial
Quadratic Polynomial:
if α and β are the zeroes of the quadratic polynomial p(x) = ax2 + bx + c,
a ≠ 0, then you know that x – α and x – β are the factors of p(x). Therefore,
ax2 + bx + c = k(x – α) (x – β), where k is a constant
= k[x2 – (α + β)x + αβ]
= kx2 – k(α + β)x + kαβ
Comparing the coefficients of x2, x and constant terms on both the sides, we get
a = k, b =– k(α + β) and c = kαβ
This gives
α + β = -b/a
αβ = c/a
Cubic Polynomial:
if α , β and γ are the zeroes of the quadratic polynomial p(x) = ax3 + bx2 + cx + d. 
How to find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
1. x2 – 2x – 8
let's factor the polynomial :
x2 – 2x – 8 = x2 -4x + 2x - 8
= x(x-4) + 2(x-4)
= (x+2)(x-4)
Equate each factor to zero , and find the value of x for each factor:
x+2 = 0 → x = -2
x-4 = 0 → x = 4
-2 and 4 are two zeros of given polynomial.
Verification the relation between zeros and cofficients :
Coefficient of x2 (a) = 1
Coefficient of x(b) = -2
constant = -8
sum of zeros = α + β = -2 + 4 = 2
-b/a = -(-2)/1 = 2
So sum of zeros = -b/a = 2
let's check product of zeros
Product of zeros = α * β = -2 * 4 = -8
Now , c/a = -8/1 = -8
so product of zeros = c/a = -8
this is the verification of relationship between zeros and the cofficients.
How to find a quadratic polynomial when the sum and product of its zeroes are given
1 . 1/4 , -1
Sum of zeros = 1/4
product of zeros = -1
α + β = -b/a = 1/4
αβ = -1
Now If a = 1 , So b = -1/4 and c = -1
Put these value in general equation of polynomial:
ax2 + bx + c = x2 + (-1/4)x + -1.
if you want that no fraction term should be not there , so simply multiply the polynomial by constant which is in denominator of any coefficient.
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