Ex 2.3

                                                  Maths NCERT Book Solutions

                                                                  Chapter 2

                                                                   Ex.2.3

Q: 1

Find the remainder when x3 + 3x2 + 3x + 1 is divided by

Answer

Let p(x) = x3 + 3x2 + 3x + 1. (i)     x + 1         Zero of x +1 is-1.         i.e. p(-1) = (- 1)3 + 3 (- 1)2 + 3 (-1) + 1 = 0         So, the remainder is 0.   (ii)                Zero of   is            (iii)     x           Zero of x is 0.           p(0) = (0)3 + 3(0)2 + 3(0) + 1 = 1           So, the remainder is 1.   (iv)     x +            Zero of x +  is:           x +  = 0  x = -            p (- ) = (- )3 + 3(- )2 + 3(- ) + 1 = - 3 + 3 2 - 3 + 1           So, the remainder is - 3 + 3 2 - 3 + 1   (v)      5 + 2x           Zero of 5+2x  is:           5 + 2x = 0  2x = - 5           i.e. x = -             Q: 2 Find the remainder when x3 - ax2 + 6x - a is divided by x - a.   Answer According to the remainder theorem, if p(x) is any polynomial of degree  1 and a is any real number, then when p(x) is divided by the linear polynomial x - a, then the remainder is p(a). Here p(x) = x3 - ax2 + 6x - a         p(a) = (a)3 - a(a)2 + 6a - a                = 5a So when x3 - ax2 + 6x - a is divided by x - a, remainder comes to 5a. OR By long division       So when x3 - ax2 + 6x - a is divided by x - a, remainder comes to 5a.   Concept Insight:  The remainder of any polynomial p(x) when divided by another polynomial (ax+b) where a and b are real numbers  is p(-b/a). Note that here -b/a  is the zero of polynomial ax+ b. This question can also be solved using long division method however it is long and time consuming. Chances of making computational error are high in that method. Q: 3 Check whether 7 + 3x is a factor of 3x3 + 7x.   Answer Zero of 7 + 3x is: 7 + 3x = 0 Therefore,  7+3x can be a factor of p(x) = 3x3 + 7x only if   Here p(x) = 3x3 + 7x    7 + 3x is not a factor of 3x3 + 7x.   OR Let us divide (3x3 + 7x) by (7 + 3x). If remainder comes out to be 0 then  7 + 3x will be a factor of 3x3 + 7x. By long division       As remainder is not zero so 7 + 3x is not a factor of 3x3 + 7x. Concept Insight: Any linear polynomial 'ax+b' where a and b are real numbers  is a factor of the polynomial p(x) iff p(-b/a) = 0 i.e  -b/a is a zero of p(x) or both the polynomials has a common zero -b/a. This question can also be solved using long division method. Do not forget to change the sign of terms while subtraction in the long division.