Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :

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 \(t^2 – 3\)

polynomial graph

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 \(x^2 + 3x + 1\)

polynomial graph

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 \(x^3 – 3x + 1\)

polynomial graph

Since the remainder is not zero, therefore, \(x^2 – 3x + 1\) is not a factor of \(x^5 – 4x^4 + x^2 + 3x + 1\).

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