ing Polynomials

K
Divide using synthetic division.
\[
\left(4 x^{3}-2 x^{2}+5 x-4\right) \div(x-2)
\]
\[
\left(4 x^{3}-2 x^{2}+5 x-4\right) \div(x-2)=\square
\]
(Simplify your answer. Do not factor.)

Step 1 ：Set up the synthetic division tableau with the coefficients of the polynomial \(4x^3 - 2x^2 + 5x - 4\) across the top, and the root of the divisor \(x - 2\) on the left. The root is the value of x that makes the divisor equal to zero, so for \(x - 2\), the root is 2.

Step 2 ：Bring down the first coefficient (4).

Step 3 ：Multiply the root (2) by the value just written below the line (4), and write the result (8) under the next coefficient (-2).

Step 4 ：Add the value above the line (-2) and the value just written below the line (8) to get 6, and write this result below the line.

Step 5 ：Repeat the previous two steps for the remaining coefficients. Multiply the root (2) by the last value written below the line (6) to get 12, write this under the next coefficient (5), add to get 17, write this below the line. Then multiply the root (2) by 17 to get 34, write this under the last coefficient (-4), add to get 30, write this below the line.

Step 6 ：The numbers on the bottom line of the tableau represent the coefficients of the quotient polynomial. The degree of the quotient is one less than the degree of the original polynomial, so the quotient is \(4x^2 + 6x + 17\). The last number on the bottom line (30) is the remainder.

Step 7 ：The final result of the division is \(4x^2 + 6x + 17 + \frac{30}{x-2}\). However, since the problem specifies not to factor, we leave the answer as \(\boxed{4x^2 + 6x + 17 + \frac{30}{x-2}}\).