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) $x^3-3x+1,x^5-4x^3+x^2+3x+1$

Answer 1

(i) Given polynomials $t^2-3,2t^4+3t^3-2t^2-9t-12$
Division of $2t^4+3t^3-2t^2-9t-12$ by $t^2-3$

Remainder is 0, hence $t^2-3$ is a factor of $2t^4+3t^3-2t^2-9t-12$

(ii) Given $x^2+3x+1,3x^4+5x^3-7x^2+2x+2$
Division of $3x^4+5x^3-7x^2+2x+2$ by $x^2+3x+1$ is

Remainder is 0, hence $x^2+3x+1$ is a factor of $3x^4+5x^3-7x^2+2x+2$

(iii) $x^3-3x+1,x^5-4x^3+x^2+3x+1$
Division of $x^5-4x^3+x^2+3x+1$ by $x^3-3x+1$ is

Remainder is 2, hence $x^3-3x+1$ is not a factor of $x^5-4x^3+x^2+3x+1$