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POLYNOMIALS
An algebraic expression of the form p(x) = a + a1x + a2x2 + a3x3+ …………….anxn, where a ≠ 0, is called a polynomial in variable x of degree n.
Here, a, 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.
• 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, 2y2– 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 αx+βx and factorise it by splitting the middle term and let factors are (x + p) and (x + q) i.e. ax2+ bx + c = 0=›(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
• Shift the constant on RHS and add square of half of the coefficient of x i.e. (b/2a)2 on both sides.
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