Divide the polynomial \$ 3x^3 - 7x^2 + 6x - 4\$ by the binomial \$(x - 2)\$.

The given polynomials are

\$ 3x^3 - 7x^2 + 6x - 4\$
\$(x - 2)\$

Identify the root of the divisor x − 2

x - 2 = 0
x = 2

Write down the coefficients of the terms of the dividend polynomial \$3x^3 - 7x^2 + 6x - 4\$ in descending order, including any missing terms with coefficient 0

3, -7, 6, -4

Set up the synthetic division, placing the root (in this case, 2) outside the division box, and write down the coefficients inside the box.

Bring down the first coefficient (3) to the bottom row of the synthetic division setup

Multiply the root (2) by the number at the bottom of the synthetic division setup (3), and write the result (6) below the next coefficient (-7).

Repeat these steps until all coefficients have been processed. The numbers in the bottom row represent the coefficients of the quotient polynomial, and the last number is the remainder.

So, the result of dividing \$ 3x^3 - 7x^2 + 6x - 4\$ by \$(x - 2)\$ is \$ 3x^2 - x + 4 \$.

This option seems incorrect because the sign of the remainder doesn’t match our calculation, which yielded a positive remainder of 4

Quotient: \$3x^2 + x - 4\$ with Remainder: -2

This option matches our calculation. The quotient has the correct coefficients for \$x^2\$ and x, and the remainder matches our result

Quotient: \$3x^2 - x + 4\$ with Remainder: 4

This option suggests that the remainder is 0, which contradicts our calculation of a remainder of 4. Additionally, the quotient doesn’t match our calculated quotient \$3x^2 + x − 2\$

Quotient: \$3x^2 - 1 \$ with Remainder: 0

This option’s remainder is incorrect; our calculation yielded a positive remainder of 4

Quotient: \$3x^2 - x + 4\$ with Remainder: -4

Option

A

Can you summarize what you’ve understood in the above steps?