1. POLYNOMIALS POLYNOMIAL – A polynomial in one variable X is an algebraic expression in X of the form NOT A POLYNOMIAL – The expression like 1  x  1,  x+2 etc are not polynomials.

2. DEGREE OF POLYNOMIAL Degree of polynomial- The highest power of x in p(x) is called the degree of the polynomial p(x). EXAMPLE – 1) F(x) = 3x +½ is a polynomial in the variable x of degree 1. 2) g(y) = 2y²  ⅜ y +7 is a polynomial in the variable y of degree 2.

3. TYPES OF POLYNOMIALS Types of polynomials are – 1] Constant polynomial 2] Linear polynomial 3] Quadratic polynomial 4] Cubic polynomial 5] Bi-quadratic polynomial

4. CONSTANT POLYNOMIAL CONSTANT POLYNOMIAL – A polynomial of degree zero is called a constant polynomial. EXAMPLE - F(x) = 7 etc. It is also called zero polynomial. The degree of the zero polynomial is not defined.

5. LINEAR POLYNOMIAL LINEAR POLYNOMIAL – A polynomial of degree 1 is called a linear polynomial. EXAMPLE- 2x  3,  3x +5 etc. The most general form of a linear polynomial is ax  b, a  0,a & b are real.

6. QUADRATIC POLYNOMIAL QUADRATIC POLYNOMIAL – A polynomial of degree 2 is called quadratic polynomial. EXAMPLE – 2x²  3x  ⅔, y²  2 etc. More generally, any quadratic polynomial in x with real coefficient is of the form ax² + bx + c, where a, b,c, are real numbers and a  0

7. CUBIC POLYNOMIALS CUBIC POLYNOMIAL – A polynomial of degree 3 is called a cubic polynomial. EXAMPLE = 2  x³, x³, etc. The most general form of a cubic polynomial with coefficients as real numbers is ax³  bx²  cx  d, a,b,c,d are reals.

8. BI QUADRATIC POLYNMIAL BI – QUADRATIC POLYNOMIAL – A fourth degree polynomial is called a biquadratic polynomial.

9. VALUE OF POLYNOMIAL If p(x) is a polynomial in x, and if k is any real constant, then the real number obtained by replacing x by k in p(x), is called the value of p(x) at k, and is denoted by p(k). For example, consider the polynomial p(x) = x²  3x  4. Then, putting x= 2 in the polynomial, we get p(2) = 2²  3  2  4 =  4. The value  6 obtained by replacing x by 2 in x²  3x  4 at x = 2. Similarly, p(0) is the value of p(x) at x = 0, which is  4.

10. ZERO OF A POLYNOMIAL A real number k is said to a zero of a polynomial p(x), if said to be a zero of a polynomial p(x), if p(k) = 0. For example, consider the polynomial p(x) = x³  3x  4. Then, p(  1) = (  1)²  (3(  1)  4 = 0 Also, p(4) = (4)²  (3  4)  4 = 0 Here,  1 and 4 are called the zeroes of the quadratic polynomial x²  3x  4.

11. HOW TO FIND THE ZERO OF A LINEAR POLYNOMIAL In general, if k is a zero of p(x) = ax  b, then p(k) = ak  b = 0, k =  b  a. So, the zero of a linear polynomial ax  b is  b  a =  ( constant term )  coefficient of x. Thus, the zero of a linear polynomial is related to its coefficients.

12. GEOMETRICAL MEANING OF THE ZEROES OF A POLYNOMIAL We know that a real number k is a zero of the polynomial p(x) if p(K) = 0. But to understand the importance of finding the zeroes of a polynomial, first we shall see the geometrical meaning of – 1) Linear polynomial. 2) Quadratic polynomial 3) Cubic polynomial

13. GEOMETRICAL MEANING OF LINEAR POLYNOMIAL For a linear polynomial ax  b, a  0, the graph of y = ax  b is a straight line. Which intersect the x axis and which intersect the x axis exactly one point (  b  2, 0 ). Therefore the linear polynomial ax  b, a  0 has exactly one zero.

14. QUADRATIC POLYNOMIAL For any quadratic polynomial ax²  bx  c, a  0, the graph of the corresponding equation y = ax²  bx  c has one of the two shapes either open upwards or open downward depending on whether a  0 or a  0.these curves are called parabolas.

15. GEOMETRICAL MEANING OF CUBIC POLYNOMIAL The zeroes of a cubic polynomial p(x) are the x coordinates of the points where the graph of y = p(x) intersect the x – axis. Also, there are at most 3 zeroes for the cubic polynomials. In fact, any polynomial of degree 3 can have at most three zeroes.

16. RELATIONSHIP BETWEEN ZEROES OF A POLYNOMIAL For a quadratic polynomial – In general, if  and  are the zeroes of a quadratic polynomial p(x) = ax²  bx  c, a  0, then we know that x   and x   are the factors of p(x). Therefore, ax²  bx  c = k ( x   ) ( x   ), Where k is a constant = k[x²  (    )x  ] = kx²  k(    ) x  k  Comparing the coefficients of x², x and constant term on both the sides. Therefore, sum of zeroes =  b  a =  (coefficients of x)  coefficient of x² Product of zeroes = c  a = constant term  coefficient of x²

17. RELATIONSHIP BETWEEN ZERO AND COEFFICIENT OF A CUBIC POLYNOMIAL In general, if , , Y are the zeroes of a cubic polynomial ax³  bx²  cx  d, then  Y =  b  a =  ( Coefficient of x² )  coefficient of x³   Y  Y  =c  a = coefficient of x  coefficient of x³  Y =  d  a =  constant term  coefficient of x³

18. DIVISION ALGORITHEM 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). This result is taken as division algorithm for polynomials.

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