- The dividend is \(x^2 + 3x + 2\) and the divisor is \(x + 1\).
- Polynomial long division confirms the quotient is \(x+2\).
- The visual representation in the problem clearly shows \(x+2\) on top of the division bar.
- The correct option is c. \(\boxed{c}\).
Explanation
- Analyzing the Division Components
The problem asks us to identify the quotient of the polynomial division \((x^2 + 3x + 2) \div (x + 1)\) based on the provided long division format.
In a polynomial division problem:
- The dividend is the polynomial being divided, which is \(x^2 + 3x + 2\).
- The divisor is the polynomial by which we divide, which is \(x + 1\).
- The quotient is the result obtained after performing the division, represented as the expression written above the horizontal bar.
- The remainder is the value left over after the final subtraction, which is \(0\) in this case.
From the provided long division diagram:
\[ \begin{array}{r} x+2 \ x + 1 \longdiv { x ^ { 2 } + 3 x + 2 } \ \frac{x^2+x}{0+2 x+2} \ \frac{2 x+2}{0} \end{array} \]
we can clearly see that the expression written on top of the division bar is \(x+2\). This is our quotient.
- Verifying the Quotient and Selecting the Correct Option
To verify this, we perform the division step-by-step:
- Divide the leading term of the dividend \(x^2\) by the leading term of the divisor \(x\): \(\frac{x^2}{x} = x\). This gives the first term of the quotient.
- Multiply the divisor \((x+1)\) by \(x\): \(x(x+1) = x^2 + x\).
- Subtract this from the dividend: \((x^2 + 3x + 2) - (x^2 + x) = 2x + 2\).
- Divide the leading term of the new remainder \(2x\) by the leading term of the divisor \(x\): \(\frac{2x}{x} = 2\). This gives the second term of the quotient.
- Multiply the divisor \((x+1)\) by \(2\): \(2(x+1) = 2x + 2\).
- Subtract this from the remainder: \((2x + 2) - (2x + 2) = 0\).
The total quotient is \(x + 2\), which matches the expression on top of the division bar. Comparing this result to the given options:
A\(x+1\)
B\(0\)
C\(x+2\)
D\(x^2+3x+2\)
The correct option is (c).
Examples
Polynomial division is useful in real-world scenarios such as optimizing engineering designs or economic models. For instance, if an engineer needs to factor out a specific component from a total structural load represented by a polynomial, they use division to simplify the expression into a more manageable form (the quotient). By identifying the quotient, they can determine the remaining factors or behavior of the system, just as we identified the factor \(x+2\) here. This helps in simplifying complex mathematical models that describe real-life phenomena.