The long division shows the first term of the quotient. Which polynomial should be...
Check the final answer first, then review the worked steps.
Answer
\(x^3 + 2x^2\)
Step-by-step solution
- Identify the dividend and divisor: The dividend is the polynomial inside the division symbol, which is $x^3 + 3x^2 + x$. The divisor is the polynomial outside the division symbol, which is $x+2$.
- Determine the first term of the quotient: To find the first term of the quotient, divide the leading term of the dividend ($x^3$) by the leading term of the divisor ($x$). This gives $\frac{x^3}{x} = x^2$. This is shown above the division bar in the problem.
- Multiply the first term of the quotient by the divisor: Multiply the first term of the quotient ($x^2$) by the entire divisor ($x+2$). This results in $x^2(x+2) = x^3 + 2x^2$.
- Subtract the result from the dividend: The polynomial that should be subtracted from the dividend first is the result from step 3. So, we subtract $x^3 + 2x^2$ from $x^3 + 3x^2 + x$. The subtraction is performed as follows: $(x^3 + 3x^2 + x) - (x^3 + 2x^2)$.