Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) t^2-3, 2t^4+3t^3-2t^2-9t-12
(ii) x^2+x+1, 3x^4+5x^3-7x^2+2x+2
(iii) x^3-3x+1, x^5-4x^3+x^2+x+1
Answers
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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Some more questions :
Apply division algorithm to find the quotient q(x) and remainder r(x) in dividing f(x) by g(x) in each of the following:
brainly.in/question/7079682
Apply division algorithm to find the quotient q(x) and remainder r(x) in dividing f(x) by g(x) in each of the following:
brainly.in/question/7060874
Answer:
(I) Yes. (ii) Yes. (iii) No.
Step-by-step explanation:
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