# 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 $$t^2 – 3$$

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 $$x^5 – 4x^4 + x^2 + 3x + 1$$.