What is the quotient of $\left(x^3+3 x^2+5 x+3\right) \div(x+1)$ ?
The objective is to find the quotient when the polynomial \(P(x) = x^3 + 3x^2 + 5x + 3\) is divided by the linear binomial \(D(x) = x + 1\). Polynomial division follows the form \(P(x) = D(x)Q(x) + R\), where \(Q(x)\) is the quotient and \(R\) is the remainder. Since the divisor is a linear polynomial of degree 1, the quotient will be a quadratic polynomial of degree 2, and the remainder will be a constant. We can determine this quotient using synthetic division, long division, or by comparing coefficients.
To divide \(x^3 + 3x^2 + 5x + 3\) by \(x + 1\), we use the root of the divisor \(x + 1 = 0\), which is \(x = -1\). We list the coefficients of the dividend: \(1\) (from \(x^3\)), \(3\) (from \(x^2\)), \(5\) (from \(x\)), and \(3\) (the constant).
Following the synthetic division procedure:
The resulting values represent the coefficients of the quotient and the remainder: \(1, 2, 3\) correspond to \(x^2, x,\) and the constant term, while the final \(0\) indicates there is no remainder. Thus, the quotient is \(x^2 + 2x + 3\).
Comparing our calculated quotient \(x^2 + 2x + 3\) with the provided options:
The calculated quotient matches option (a). Therefore, the correct quotient is \(x^2 + 2x + 3\).
Polynomial division is a fundamental technique used in algebra to simplify complex algebraic expressions and solve equations. For example, if you are designing a digital signal processing filter and need to analyze its transfer function, you often divide polynomials to isolate specific components of the signal behavior. Mastering this process allows you to break down high-degree polynomials into manageable parts, much like factoring large numbers into prime factors to understand their underlying structure.