Question

Check in which case the first polynomial is a factor of the second polynomial by dividing
the second polynomial by the first polynomial :
(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12
(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2
(iii) x³ – 3x + 1, x5 – 4x³ + x² + 3x + 1

Answer 1

(i)
t² - 3)2t⁴ + 3t³ - 2t² - 9t - 12 ( 2t² + 3t + 4
2t⁴ - 6t²
----------------
3t³ + 4t² -9t
3t³ - 9t
--------------
4t² - 12
4t² - 12
-----------------
0
here we get remainder = 0
hence, t² - 3 is a factor of 2t⁴ + 3t³ - 2t² - 9t -12

ii)
x² + 3x + 1)3x⁴ + 5x³ - 7x² + 2x + 2(3x² - 4x + 2
3x⁴ + 9x³ + 3x²
--------------------
-4x³ -10x² + 2x
-4x³ -12x² - 4x
------------------------
2x² + 6x + 2
2x² + 6x + 2
------------------
0
here remainder = 0
so, x² + 3x + 1 is a factor of 3x⁴ + 5x³ - 7x² + 2x + 2 .

iii)
x³ – 3x + 1) x5 – 4x³ + x² + 3x + 1(x² -1
x5 - 3x³ + x²
-------------------
-x³ + 3x + 1
-x³ + 3x - 1
------------------
+2
hence, remainder = 2 ≠ 0
so, x - 3x + 1 is not a factor of x5 - 4x³ + x² + 3x + 1

Answer 2

SOLUTION:

Divide the second polynomial by first polynomial.

(i)If remainder is zero ,then first polynomial is a factor of the second polynomial.

(ii) If remainder is not zero, then first polynomial is not a factor of second polynomial.

(i) The division process is :

t² - 3)2t⁴ + 3t³ - 2t² - 9t - 12 ( 2t² + 3t + 4

        2t⁴        - 6t²  

       (-)         (+)

       ----------------

             3t³ + 4t² -9t

             3t³        - 9t

            (-)         (+)

             --------------

                    4t² - 12  

                   4t² - 12

                  (-)   (+)

             -----------------

                      0  

Here, the remainder is 0.

Hence, t² - 3 is a factor of 2t⁴ + 3t³ - 2t² - 9t - 12.

ii)  

x² + 3x + 1)3x⁴ + 5x³ - 7x² + 2x + 2(3x² - 4x + 2

                 3x⁴ + 9x³ + 3x²  

                (-)   (-)      (-)

               --------------------

                      - 4x³ -10x² + 2x  

                      - 4x³ -12x² - 4x  

                      (+)  (+)     (+)

                     ------------------------

                             2x² + 6x + 2  

                             2x² + 6x + 2  

                             (-)  (-)     (-)

                          ------------------

                                   0  

Here,the remainder is 0 .

Hence, x² + 3x + 1 is a factor of 3x⁴ + 5x³ - 7x² + 2x + 2 .

iii)

x³ – 3x + 1) x^5 – 4x³ + x² + 3x + 1(x² -1

                   x^5 - 3x³ + x²  

                  (-)    (+)     (-)

                  -------------------

                         -x³ + 3x + 1

                          -x³ + 3x - 1

                         (+)  (-)     (+)

                        ------------------

.                                       +2  

Here, remainder is 2 ≠ 0

Hence, x³ - 3x + 1 is not a factor of x^5 - 4x³ + x² + 3x + 1

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