LF-ASSESSMENT 6. Divide (a)(x^3-3x^2-7x+6)div (x-2) and (b)(2x^3-16x+6)div (x+3) using synthetic division nolvnomials closed under division? Justify your answer.

To divide the given polynomials using synthetic division, we will follow the steps below:<br /><br />(a) $(x^{3}-3x^{2}-7x+6)\div (x-2)$<br /><br />Step 1: Write down the coefficients of the dividend polynomial: $1, -3, -7, 6$.<br /><br />Step 2: Write down the zero of the divisor polynomial, which is $2$.<br /><br />Step 3: Bring down the first coefficient, which is $1$.<br /><br />Step 4: Multiply the zero of the divisor by the value just brought down, and write the result under the next coefficient.<br /><br />Step 5: Add the values in the second column and write the result below the line.<br /><br />Step 6: Repeat steps 4 and 5 until all coefficients have been processed.<br /><br />The synthetic division table for this problem is as follows:<br /><br />```<br /> 2 | 1 -3 -7 6<br /> | 2 -2 -18<br /> ----------------<br /> 1 -1 -9 -12<br />```<br /><br />The quotient is $x^2 - x - 9$ and the remainder is $-12$.<br /><br />(b) $(2x^{3}-16x+6)\div (x+3)$<br /><br />Step 1: Write down the coefficients of the dividend polynomial: $2, 0, -16, 6$.<br /><br />Step 2: Write down the zero of the divisor polynomial, which is $-3$.<br /><br />Step 3: Bring down the first coefficient, which is $2$.<br /><br />Step 4: Multiply the zero of the divisor by the value just brought down, and write the result under the next coefficient.<br /><br />Step 5: Add the values in the second column and write the result below the line.<br /><br />Step 6: Repeat steps 4 and 5 until all coefficients have been processed.<br /><br />The synthetic division table for this problem is as follows:<br /><br />```<br /> -3 | 2 0 -16 6<br /> | -6 18 -6<br /> ----------------<br /> 2 -6 2 0<br />```<br /><br />The quotient is $2x^2 - 6x + 2$ and the remainder is $0$.<br /><br />Therefore, the answers are:<br />(a) $x^2 - x - 9$<br />(b) $2x^2 - 6x + 2$<br /><br />Polynomials are closed under division, meaning that the division of two polynomials always results in another polynomial. In this case, both $(x^{3}-3x^{2}-7x+6)\div (x-2)$ and $(2x^{3}-16x+6)\div (x+3)$ resulted in polynomials as the quotients.