Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{3 x^{3}+2 x-5}{x-1} $$

Step-by-Step Solution

Answer

The quotient is \(3x^2 + 3x + 5\) with a remainder of \(0\).

Step 1: Set Up Synthetic Division

Identify the divisor and set up the synthetic division by noting the root of the divisor. Since our divisor is \(x - 1\), the root is \(1\). Write the coefficients of the dividend \(3x^3 + 0x^2 + 2x - 5\) in order, which gives us \([3, 0, 2, -5]\).

Step 2: Begin Synthetic Division

Bring down the first coefficient, \(3\), directly below the line to the next row. This is the starting point for synthetic division.

Step 3: Multiply and Add

Multiply the root, \(1\), by the \(3\) that was just brought down, resulting in \(3\). Add this to the next coefficient \(0\) from the dividend, getting \(3\). Write this result immediately below and move to the next coefficient.

Step 4: Repeat Multiply and Add

Again, multiply \(1\) by the last result \(3\), which is \(3\), and add this to the next coefficient \(2\) from the dividend, resulting in \(5\). Write this result below and move to the final coefficient.

Step 5: Final Multiply and Add

Multiply \(1\) by \(5\), giving \(5\), and add it to the last coefficient \(-5\) from the dividend, resulting in \(0\). This is the remainder of the division.

Step 6: Interpret the Results

The results from the synthetic division give us the coefficients of the quotient and the remainder. The quotient is \(3x^2 + 3x + 5\) and the remainder is \(0\). Since the remainder is zero, \(x - 1\) is a factor of the dividend.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to the way we'd divide numbers. When you have a polynomial called the dividend, like \(3x^3 + 2x - 5\), and you want to divide it by another polynomial, called the divisor \(x-1\), you would use either long division similar to arithmetic or synthetic division, which is a shorter process.

**Why Use Polynomial Division?**

• To simplify complex algebraic expressions.
• To find out if a polynomial divides evenly into another; this helps identify factors.
• To break down higher degree polynomials into lower degree ones.

In our exercise, synthetic division was used. It's a streamlined process, ideal for cases where the divisor is a linear polynomial with a leading coefficient of 1, making it faster and often less error-prone than traditional long division.

Remainder Theorem

The Remainder Theorem is a handy concept in algebra, especially when dealing with polynomials. It states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder of this division is \(f(c)\). This tells us that plugging in the value of \(c\) directly into the polynomial gives the remainder!

In our synthetic division example, when dividing \(3x^3 + 2x - 5\) by \(x-1\), the remainder was calculated to be 0. According to the Remainder Theorem, this means that substituting \(x = 1\) into the polynomial, the result is 0. Therefore:
\[f(1) = 3(1)^3 + 0(1)^2 + 2(1) - 5 = 0\]
This is confirmation that \(x-1\) is a factor of the polynomial. Understanding the Remainder Theorem can thus streamline checks for factors and roots without completing the entire division each time.

Quotients in Algebra

Quotients in the context of algebra are the result you get when dividing one polynomial by another. When you perform division, you're essentially unpacking a bigger polynomial into smaller, more manageable parts. This is useful in simplifying expressions or solving equations.

**Finding the Quotient:**

• Set up using either synthetic or traditional division.
• Solve step by step, interpreting results at each stage.
• For polynomials, like \(3x^3 + 2x - 5\) divided by \(x-1\), the quotient found was \(3x^2 + 3x + 5\).

Understanding quotients is crucial as they reveal the polynomial's structure after division. In our case, it demonstrated that \(x-1\) divides evenly into the given polynomial, leaving no remainder. Recognizing these patterns can simplify factoring and further manipulation of polynomials in algebra.