\begin{tabular}{|l} I can divide polynomials using the pox meurivu. \ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \ $$ \left(4 x^{3}+x-2 ight) \div i(2 x-1) $$ \ \hline\end{tabular}

Question

\begin{tabular}{|l} I can divide polynomials using the pox meurivu. \ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \ $$ \left(4 x^{3}+x-2ight) \div i(2 x-1) $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

$$
\textbf{Step 1: Write the dividend in full form.}
$$
Write the dividend $$ 4x^3+x-2 $$ with all degrees:
$$
4x^3+0x^2+x-2.
$$

$$
\textbf{Step 2: Divide the leading term of the dividend by the leading term of the divisor.}
$$
Divisor: $$2x-1.$$  
Divide $$4x^3$$ by $$2x$$:
$$
\frac{4x^3}{2x} = 2x^2.
$$
So, the first term of the quotient is $$2x^2$$.

$$
\textbf{Step 3: Multiply and subtract.}
$$
Multiply the divisor $$2x-1$$ by $$2x^2$$:
$$
2x^2\cdot(2x-1)=4x^3-2x^2.
$$
Subtract this from the dividend:
$$
\begin{array}{rcl}
(4x^3+0x^2+x-2) - (4x^3-2x^2) &=& (4x^3-4x^3) + (0x^2+2x^2) + x-2 \
&=& 2x^2+x-2.
\end{array}
$$

$$
\textbf{Step 4: Repeat the process with the new polynomial.}
$$
Divide the new leading term $$2x^2$$ by $$2x$$:
$$
\frac{2x^2}{2x} = x.
$$
So, the next term in the quotient is $$x$$.

Multiply the divisor by $$x$$:
$$
x\cdot(2x-1)=2x^2 -x.
$$
Subtract:
$$
\begin{array}{rcl}
(2x^2+x-2) - (2x^2-x) &=& (2x^2-2x^2) + (x+x)-2 \
&=& 2x-2.
\end{array}
$$

$$
\textbf{Step 5: Continue the division.}
$$
Now, divide the leading term $$2x$$ by $$2x$$:
$$
\frac{2x}{2x}=1.
$$
The next term in the quotient is $$1$$.

Multiply the divisor by $$1$$:
$$
1\cdot(2x-1)=2x-1.
$$
Subtract:
$$
\begin{array}{rcl}
(2x-2) - (2x-1) &=& 2x-2x -2+1 \
&=& -1.
\end{array}
$$
The remainder is $$-1$$.

$$
\textbf{Step 6: Write the final result.}
$$
The quotient is:
$$
2x^2 + x+ 1,
$$
with a remainder of $$-1$$. Therefore, we can express the division as:
$$
\frac{4x^3+x-2}{2x-1}=2x^2+x+1-\frac{1}{2x-1}.
$$
  
This is the final answer.
The division of $$4x^3 + x - 2$$ by $$2x - 1$$ results in a quotient of $$2x^2 + x + 1$$ with a remainder of $$-1$$. So, the final expression is: $$ 2x^2 + x + 1 - \frac{1}{2x - 1} $$

\begin{tabular}{|l} I can divide polynomials using the pox meurivu. \\ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \\ \( \left(4 x^{3}+x-2\right) \div i(2 x-1) \) \\ \hline\end{tabular}

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