Question : 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) \(x^2 + 3x + 1\) ; \(3x^4 + 5x^3 – 7x^2 + 2x + 2\)
(iii) \(x^3 – 3x + 1\) ; \(x^5 – 4x^4 + x^2 + 3x + 1\)
Solution :
(i) Let us divide \(2t^4 + 3t^3 – 2t^2 – 9t – 12\) by \(t^2 – 3\)
Since the remainder is zero, therefore, \(t^2 – 3\) is a factor of \(2t^4 + 3t^3 – 2t^2 – 9t – 12\)
(ii) Let us divide \(3x^4 + 5x^3 – 7x^2 + 2x + 2\) by \(x^2 + 3x + 1\)
Since the remainder is zero, therefore, \(x^2 + 3x + 1\) is a factor of \(3x^4 + 5x^3 – 7x^2 + 2x + 2\)
(iii) Let us divide \(x^5 – 4x^4 + x^2 + 3x + 1\) by \(x^3 – 3x + 1\)
Since the remainder is not zero, therefore, \(x^2 – 3x + 1\) is not a factor of \(x^5 – 4x^4 + x^2 + 3x + 1\).
