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POLYNOMIALS
An algebraic expression of the form p(x) = a0 + a1x + a2x2 + a3x3+ …………….anxn, where a ≠ 0, is called a polynomial in variable x of degree n.
Here, a0, a1, a2, a3, ………,an are real numbers and each power of x is a non-negative integer. e.g. 3x2 – 5x + 2 is a polynomial of degree 2.
3 2 x + is not a polynomial.
Ø 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). For example, 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,
A polynomial of degree 0 is called a constant polynomial.
A polynomial p(x) = ax + b of degree 1 is called a linear polynomial.
A polynomial p(x) = ax2 + bx + c of degree 2 is called a quadratic polynomial.
A polynomial p(x) = ax3 + bx2 + cx + d of degree 3 is called a cubic polynomial.
A polynomial p(x) = ax4+ bx3 + cx2 + dx + e of degree 4 is called a bi-quadratic polynomial.
QUADRATIC EQUATION
A polynomial p(x) = ax2 + bx + c of degree 2 is called a quadratic polynomial, then p(x) = 0 is
known as quadratic equation.
e.g. 2x2 – 3x + 2 = 0, x2 + 5x + 6 = 0 are quadratic equations.
METHODS TO FIND THE SOLUTION OF QUADRATIC EQUATIONS
Three methods to find the solution of quadratic equation:
1. Factorisation method
2. Method of completing the square
3. Quadratic formula method
FACTORISATION METHOD
Steps to find the solution of given quadratic equation by factorisation
Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0.
Find two numbers a and b such that sum of a and b is equal to b and product of a and b is equal to ac.
Write the middle term bx as a b x x + and factorise it by splitting the middle term and let factors are (x + p) and (x + q) i.e. ax2
+ bx + c = 0 Þ (x + p)(x + q) = 0
Now equate reach factor to zero and find the values of x.
These values of x are the required roots/solutions of the given quadratic equation.
METHOD OF COMPLETING THE SQUARE
Steps to find the solution of given quadratic equation by Method of completing the square:
Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0.
Make coefficient of x2 unity by dividing all by a then we get
x2+b/ax+c/a=0
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