# CBSE Worksheet for chapter-2 Polynomial class 10

## Summary

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.

### IMPORTANT FORMULAS

• (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

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

#### OBJECTIVE

Q1. If 4x4-3x3-3x2+x+7 is divided by 1-2x then remainder will be:

1. 57/8
2. (-59)/8
3. 55/8
4. (-55)/8

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:

1. x2 + 6x + 4
2. 2x2 + 6x + 4
3. 2x2 + 6x - 4
4. x2 + 6x - 4

Q3. Graph of a quadratic equation is always;

1. straight line
2. circle
3. parabola
4. hyperbola

Q4. The graph of a polynomial y = x3 - x2 + x is always passing through the point

1. (0,0)
2. (3,2)
3. (1,-2)
4. all of these

Q5. Which of the following touches X-axis?

1. x2 - 2x + 4
2. 3x2 - 6x + 1
3. 4x2 - 16x + 9
4. 25x2 - 20x + 4

#### SUBJECTIVE

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?

Objective:

1. 2
2. 2
3. 3
4. 1
5. 4
6. 1

Subjective:

Q1. 3,-3

Q2. k(x2 - 6x + 4)

Q3. x2 - x + 1

Q4.c = - 48

Q5.2x - 3

Q6. k = -1

Q7. -1

Q8. k = - 21