The long division shows the first term of the quotient. Which polynomial should be...
Check the final answer first, then review the worked steps.
Problem
The long division shows the first term of the quotient. Which polynomial should be subtracted from the dividend first?
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: $(x^3 + 3x^2 + x) - (x^3 + 2x^2)$.