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An algebraic expression f(x) of the form f(x)= a0x2+…….+an xn ,where a0, a1, a2……. an 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 a3+b3+c3 = 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
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.
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 4x4-3x3-3x2+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 = x3 - x2 + x is always passing through the point
Q5. Which of the following touches X-axis?
Q1. Find the zeros of quadratic polynomial p(x) = 4x2 + 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 x3- 3x2 - 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 x3-12x2- 44 - c.If α ,β,⅟ are in A.P, find values of c.
Q5. What must be subtracted fromx4 + 2x3 - 13x2 - 12x + 21 so that the result is exactly divisible by x2 - 4x + 3
Q6. If α ,β are zero of quadratic polynomial kx2 + 4x + 4,find the values of k such that (α + β)2 -2 αβ =24.
Q7. If α, β are zero of quadratic polynomial 2y2 + 7y + 5,write the values of α + β+ αβ.
Q8. For what values of k, is 3 a zero of y the polynomial 2x2 + x + k?
Q2. k(x2 - 6x + 4)
Q3. x2 - x + 1
Q4.c = - 48
Q5.2x - 3
Q6. k = -1
Q8. k = - 21