What is the quotient of $\left(x^3+6 x^2+11 x+6\right) \div\left(x^2+4 x+3\right)$?
The objective is to find the quotient of the polynomial division \((x^3 + 6x^2 + 11x + 6) \div (x^2 + 4x + 3)\). To simplify this division, we can first factor both the dividend and the divisor.
For the dividend \(p(x) = x^3 + 6x^2 + 11x + 6\), we test simple integer roots using the Rational Root Theorem. Checking \(x = -1\), we find \(p(-1) = (-1)^3 + 6(-1)^2 + 11(-1) + 6 = -1 + 6 - 11 + 6 = 0\), so \((x + 1)\) is a factor. Dividing \(x^3 + 6x^2 + 11x + 6\) by \((x + 1)\), we obtain \(x^2 + 5x + 6\), which further factors into \((x + 2)(x + 3)\). Thus, the complete factorization is \(p(x) = (x + 1)(x + 2)(x + 3)\).
For the divisor \(d(x) = x^2 + 4x + 3\), we look for two numbers that multiply to \(3\) and add to \(4\). These numbers are \(1\) and \(3\), giving us the factors \((x + 1)(x + 3)\).
Now that we have the factored forms of the polynomials, the division \(\frac{p(x)}{d(x)}\) becomes:
\[\frac{p(x)}{d(x)} = \frac{(x + 1)(x + 2)(x + 3)}{(x + 1)(x + 3)}\]
Assuming \(x
eq -1\) and \(x
eq -3\), we can cancel the common terms \((x + 1)\) and \((x + 3)\) in the numerator and denominator:
\[\frac{(x + 1)(x + 2)(x + 3)}{(x + 1)(x + 3)} = x + 2\]
Therefore, the quotient of the division is \(x + 2\), which corresponds to option (b).
To confirm the result, we can perform long division or multiply the quotient by the divisor to see if it equals the dividend:
\[(x + 2)(x^2 + 4x + 3) = x(x^2 + 4x + 3) + 2(x^2 + 4x + 3)\]
\[= (x^3 + 4x^2 + 3x) + (2x^2 + 8x + 6)\]
\[= x^3 + (4x^2 + 2x^2) + (3x + 8x) + 6\]
\[= x^3 + 6x^2 + 11x + 6\]
The result matches the original dividend, verifying that the quotient is indeed \(x + 2\). The correct option is (b). The final answer is \(\boxed{x+2}\).
Polynomial division is essential in engineering and physics, particularly when analyzing transfer functions or electrical circuits. For instance, when designing filters for audio systems, engineers divide the output polynomial signal by the input polynomial signal (a process called deconvolution) to determine the system's characteristic behavior or its impulse response. Similarly, solving such divisions helps in simplifying complex equations that model structural vibrations or thermal diffusion in materials.