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

$x^{2} + 3 x + 1, 3 x^{4} + 5 x^{3} - 7 x^{2} + 2 x + 2$

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Solution

$\begin{matrix} 3 x^{2} - 4 x + 2 x^{2} + 3 x + 1 & 3 x^{4} + 5 x^{3} - 7 x^{2} + 2 x + 2 3 x^{4} + 9 x^{3} + 3 x^{2} - - - ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - 4 x^{3} - 10 x^{2} + 2 x + 2 - 4 x^{3} - 12 x^{2} - 4 x ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 x^{2} + 6 x + 2 2 x^{2} + 6 x + 2 - ----------------------- - 0 \end{matrix}$

Since the remainder is 0,
$x^{2} + 3 x + 1$ exactly divides $3 x^{4} + 5 x^{3} - 7 x^{2} + 2 x + 2$ leaving no remainder.

Hence, it is factor of $3 x^{4} + 5 x^{3} - 7 x^{2} + 2 x + 2.$

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Question 2 (ii)
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

$x^{2} + 3 x + 1, 3 x^{4} + 5 x^{3} - 7 x^{2} + 2 x + 2$

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

t^{2} - 3, 2 t^{4} + 3 t^{3} - 2 t^{2} - 9 t - 12

Question 2(i)
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
$t^{2} - 3, 2 t^{4} + 3 t^{3} 2 t^{2} - 9 t - 12$

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Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: x2+3x+1,3x4+5x3−7x2+2x+2

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

$x^{2} + 3 x + 1, 3 x^{4} + 5 x^{3} - 7 x^{2} + 2 x + 2$