NCERT Class 9 Solutions: Polynomials (Chapter 2) Exercise 2.3

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

  1. x+1

  2. x12

  3. x

  4. x+π

  5. 5+2x

Solution:

How does how step of long division of polynomail work

Understaning the Division of Polynomials

How does how step of long division of polynomail work

  1. x+1

    By long division,

    x+1)x2+2x+1x3+3x2+3x+1x3+x22x2+3x+12x2+2xx+1x+10

    Therefore, the remainder is 0.

  2. x12

    By long division,

    x12)x2+72x+194x3+3x2+3x+1x3x22+72x2+3x+172x274x+194x+1194x198+278

    Therefore, the remainder is 278

  3. x

    By long division,

    x)x2+3x+3x3+3x2+3x+1x33x2+3x+13x23x+13x1

    Therefore, the remainder is 1

  4. x+π

    By long division,

    x+π)x2+(3π)x+(33π+π2)x3+3x2+3x+1x3+πx2(3π)x2+3x+1(3π)x2+(3π)πx[33π+π2]x+1[33π+π2]x+[33π+π2]π[13π+3π2π3]

    Therefore, the remainder is [13π+3π2π3]

  5. 5+2x

    By long division,

    2x+5)x22+x4+78x3+3x2+3x+1x3+52x2x22+3x+1x22+5x474x+174x+358278

    Therefore, the remainder is 278

Q-2 Find the remainder when x3ax2+6xa is divided by xa .

Explaining the division of polynomial with an example

Demonstrating Division of Polynomial

Explaining the division of polynomial with an example

Solution:

By long Division,

xa)x2+6x3ax2+6xax3ax2+6xa6x6a+5a

Therefore, remainder obtained is 5a when x3ax2+6xa is divided by xa

Q-3 Check whether 7+3x is a factor of 3x3+7x

Solution:

  • We have to 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,

    3x7)x273x+7093x3+0x2+7x3x3+7x27x2+7x7x2493x++703x703x+49094909

    As remainder is not zero so 7+3x is not a factor of 3x3+7x

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