Question

Using long division what is the quotient of this expression 3x4-2x3-x-4 x2+2

Answer 1

have to divide 3x⁴ + 2x² - 3 by (x + 1) to obtain the remainder and quotient. therefore remainder = 2 and quotient = 3x³ - 3x² + 5x - 5.

Answer 2

The quotient of the expression is
`3 x^2-2 x-6+(3 x+8)/(x^2+2)`. Hence, option (a) is correct.

Step 1: Set up the long division.

`$$\begin{aligned}x^2+2)\overline {3 x^4-2 x^2-z-4} \\3 x^4+6 x^2 \\------- \\-8 x^2-z-4 \\-8 x^2-16 \\------ \\15-z-4\end{aligned}$$`

Step 2: Divide the first term of the dividend by the first term of the divisor.

The leading term of the dividend is
`3x^4` , and the leading term of the divisor is
`x^2`. Dividing
`3x^4` by
`x^2`, gives 3x

Step 3: Multiply the divisor by the quotient.

`$$\begin{aligned}& x^2+2)\overline{ 3 x^4-2 x^2-z-4} \\& 3 x^4+6 x^2 \\&------ \\&-8 x^2-z-4 \\&-8 x^2-16 \\&----- \\& 15-z-4\end{aligned}$$`

Step 4: Subtract the product from the dividend.

Subtracting
`$3 x^4+6 x^2$` from
`$3 x^4-2 x^2-z-4$` gives
`$-8 x^2-z-4$`.

Step 5: Bring down the next term of the dividend.

The next term of the dividend is -z-4.

Step 6: Repeat steps 2-5 until the remainder is either zero or a polynomial of lower degree than the divisor.

Dividing
`$-8 x^2-z-4$` by
`$x^2+2$` gives
`$-8 x^2-16+15-z-4$`. Since the remainder is 15-z-4, which is a polynomial of lower degree than the divisor, the division is complete.

Step 7: Write the quotient and remainder.

The quotient is
`$3 x^2-2 x-6$`, and the remainder is 15-z-4. Therefore, the final answer is
`3 x^2-2 x-6+(3 x+8)/(x^2+2).`