Mathematic Worksheet - 2
For other CBSE Worksheet for class 10 Mathematic check out main page of Entrancei.
An algebraic expression f(x) of the form f(x)= a0 + a1x +a2x2+…….+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.
(x+a)2 =x2 +2ax+a2
(x-a)2 =x2 – 2ax +a2
x2-a2 =(x+a) (x-a)
x3+a3=(x+a) (x2-ax+a2)=(x+a)3 –3xa(x+a)
x3-a3=(x-a) (x2+ax+a2)=(x+a)3 +3xa(x-a)
(a + b + c) 2=a2+b2+c2+2ab+2bc+2ac
(a + b) 3= a3+b3+3ab (a + b)
(a-b) 3= a3-b3-3ab (a-b)
a3+b3+c3-3abc = (a +b +c) (a2+b2+c2-ab-bc-ca)
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
1) If 4x4-3x3-3x2+x+7 is divided by 1-2x then remainder will be:
2) 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:
3) Graph of a quadratic equation is always;
a) straight line
4) The graph of a polynomial y=x3-x2+x is always passing through the point
d) all of these
5) Which of the following touches X-axis?
1) Find the zeros of quadratic polynomial p(x)=4x2+24x+36 and verify the relationship between the zeros and their coefficient.
2) Find the quadratic polynomial whose zeros are 3+ and 3-
3) 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).
4) α, β, ⅟ are zeros of cubic polynomial x3-12x2-44-c.If α ,β,⅟ are in A.P, find values of c.
5) What must be subtracted fromx4+2x3-13x2-12x+21 so that the result is exactly divisible by x2-4x+3
6) If α ,β are zero of quadratic polynomial kx2+4x+4,find the values of k such that (α + β)2 -2 αβ =24.
7) If α, β are zero of quadratic polynomial 2y2+7y+5,write the values of α + β+ αβ.
8) For what values of k, is 3 a zero of y the polynomial 2x2+x+k?
6) k= ,-1