An algebraic expression f(x) of the form f(x)= a_{0}x^{2}+…….+a_{n }x^{n },where a_{0}, a_{1,} a_{2}……. a_{n } are real numbers and all the index of x are non negative integers is called polynomial in x and the highest index n is called as the degree of polynomial.
SPECIAL CASE : If a+ b +c=0 then a^{3}+b^{3}+c^{3 }= 3abc
ZEROS OR ROOTS OF A POLYNOMIAL
A real number ‘a’ is a zero of a polynomial f(x),if f(a)=0.Here ‘a’ is called a root of the equation f(x) = 0
FACTOR THEOREM
Let p(x) be a polynomial of degree greater than or equal to ‘a’, be a real number such that p(a)=0,then (x-a) is a factor of p(x),then p(a)=0.
REMAINDER THEOREM
Let p(x) be any polynomial of degree greater than or equal to one and ‘a’ be any real number. If p(x)is a dividend by (x-a),then remainder is equal to p(a).let q(x) be the quotient and r(x) be the remainder when p(x) is divided by (x-a),then
Dividend=Divisor x Quotient + Remainder
Q1. If 4x^{4}-3x^{3}-3x^{2}+x+7 is divided by 1-2x then remainder will be:
Q2. A quadratic polynomial is exactly divisible by(x+1) and (x+2) and leaves the remainder 4 after division by (x+3) then that polynomial is:
Q3. Graph of a quadratic equation is always;
Q4. The graph of a polynomial y = x^{3 }- x^{2 }+ x is always passing through the point
Q5. Which of the following touches X-axis?
Q1. Find the zeros of quadratic polynomial p(x) = 4x^{2 }+ 24x + 36 and verify the relationship between the zeros and their coefficient.
Q2. Find the quadratic polynomial whose zeros are 3+ and 3-
Q3. On dividing x^{3}- 3x^{2 }- x - 2 by a polynomial g(x), The quotient and remainder were x - 2 and -2x + 4 respectively. Find g(x).
Q4. α, β, ⅟ are zeros of cubic polynomial x^{3}-12x^{2}- 44 - c.If α ,β,⅟ are in A.P, find values of c.
Q5. What must be subtracted fromx^{4} + 2x^{3 }- 13x^{2 }- 12x + 21 so that the result is exactly divisible by x^{2} - 4x + 3
Q6. If α ,β are zero of quadratic polynomial kx^{2} + 4x + 4,find the values of k such that (α + β)^{2 }-2 αβ =24.
Q7. If α, β are zero of quadratic polynomial 2y^{2} + 7y + 5,write the values of α + β+ αβ.
Q8. For what values of k, is 3 a zero of y the polynomial 2x^{2} + x + k?
Objective:
Subjective:
Q1. 3,-3
Q2. k(x^{2} - 6x + 4)
Q3. x^{2} - x + 1
Q4.c = - 48
Q5.2x - 3
Q6. k = -1
Q7. -1
Q8. k = - 21