Problem 329 - Precalculus

Question

Use synthetic division to find the quotient $$\left(6 x^{3}-x^{2}+5 x+2\right) \div(3 x+1)$$

Step-by-Step Solution

Answer

The quotient is \(6x^2 - 3x + 6\) with no remainder.

Step 1: Write the Dividend and Divisor

Identify the dividend and the divisor in the given problem. The dividend is \(6x^3 - x^2 + 5x + 2\), and the divisor is \(3x + 1\). For synthetic division, we'll treat the divisor as \(3x + 1 = 0\) and solve for \(x\) to get \(x = -\frac{1}{3}\).

Step 2: Set up Synthetic Division

Write down the coefficients of the dividend: \(6, -1, 5, 2\). The divisor root or the number we found \(-\frac{1}{3}\) will be used in synthetic division.

Step 3: Perform Synthetic Division

1. Bring down the leading coefficient \(6\).2. Multiply \(6\) by \(-\frac{1}{3}\) and write it under the next coefficient \(-1\), then add: \(-1 + (-2) = -3\).3. Multiply \(-3\) by \(-\frac{1}{3}\), write the result \(1\) under the next coefficient \(5\), and add: \(5 + 1 = 6\).4. Multiply \(6\) by \(-\frac{1}{3}\), and write the result \(-2\) under \(2\), and add: \(2 - 2 = 0\).

Step 4: Interpret the Result

The quotient is formed from the results excluding the remainder. The final row of numbers from the synthetic division represents: \(6x^2 - 3x + 6\). Since the remainder is \(0\), this means the division is exact and the divisor divides the dividend perfectly.

Key Concepts

polynomial division

When dealing with algebraic expressions, polynomial division is an essential technique for dividing two polynomials. It's similar to long division with numbers but applied to expressions with variables.
Polynomial division is often used to simplify expressions or solve equations involving polynomials. It's the process of dividing a polynomial, known as the dividend, by another polynomial, referred to as the divisor. The result will include a quotient (which is another polynomial) and sometimes a remainder.
In cases where the divisor is a linear polynomial, synthetic division can be a faster and more efficient method compared to traditional long division. Synthetic division is beneficial as it simplifies polynomial division into working with numbers and basic arithmetic, which makes it easier for simpler cases.
Understanding this concept not only helps solve polynomial equations but also provides a basis for exploring more complex algebraic techniques.

remainder theorem

The Remainder Theorem provides a quick method for finding the remainder of a polynomial division without having to perform the entire division. It states that if you divide a polynomial \(f(x)\) by a linear divisor \((x - c)\), the remainder of this division is simply \(f(c)\).
This theorem is particularly useful because it helps to quickly verify if \((x - c)\) is indeed a factor of \(f(x)\). If plugging \(c\) into the polynomial results in a remainder of zero, then \((x - c)\) is a factor.
In synthetic division, you often encounter the Remainder Theorem because if the remainder is zero, it confirms that the divisor divides the polynomial perfectly, as seen in the given exercise. It simplifies problems and supports determining factors and roots of polynomials effectively.

dividing polynomials

Dividing polynomials involves breaking a polynomial into simpler parts by a divisor to obtain a quotient and possibly a remainder.
There are two primary methods for dividing polynomials:

• Long Division: Similar to numeric long division, it's often used for dividing a polynomial by another polynomial of higher or equal degree.
• Synthetic Division: A streamlined method specifically for divisors in the form of \((x - c)\), making it quicker than long division, especially for higher-degree polynomials with lower-degree simplicity.

In the given exercise, synthetic division simplifies the division of a third-degree polynomial by a first-degree linear divisor. It's a practical application to discover the quotient and remainder in polynomial division.

roots of equations

Roots of an equation are the values of the variable that satisfy the equation, making it true. For polynomial equations, these are the values of \(x\) that make the polynomial equal to zero.
Determining roots is crucial in solving polynomial equations. They indicate where the polynomial will intersect the x-axis on a graph. Each root corresponds to a factor of the polynomial.In polynomial division, especially when using synthetic division, we often search for these roots using the divisor in the form \((x - c)\). The roots can confirm our division results: if a polynomial could be perfectly divided by \((x - c)\), then \(c\) is one of its roots, as demonstrated in the division without a remainder.
Finding roots not only solves equations but also aids in factoring and analyzing polynomial functions by determining their behavior across different x-values.