Problem 315 - Precalculus

Question

Use synthetic division to find the quotient $$\left(3 x^{3}-2 x^{2}+x-4\right) \div(x+3)$$

Step-by-Step Solution

Answer

The quotient is \(3x^2 - 11x + 34\) with a remainder of \(-106\).

Step 1: Set Up the Synthetic Division

First, identify the coefficients of the polynomial \(3x^3 - 2x^2 + x - 4\), which are \([3, -2, 1, -4]\). Since we are dividing by \((x + 3)\), use \(-3\) as the divisor.

Step 2: Bring Down the Leading Coefficient

Write down the leading coefficient \(3\) directly under the division line. This will be the first number in our solution row for synthetic division.

Step 3: Multiply and Add Down

Multiply \(3\) by \(-3\) (our synthetic divisor) to get \(-9\). Place \(-9\) under the next coefficient \(-2\). Add \(-2\) and \(-9\) to get \(-11\).

Step 4: Repeat Multiplying and Adding

Multiply \(-11\) by \(-3\) to get \(33\). Place \(33\) under the next coefficient \(1\). Add \(1\) and \(33\) to get \(34\).

Step 5: Final Multiply and Add

Multiply \(34\) by \(-3\) to get \(-102\). Place \(-102\) under the last coefficient \(-4\). Add \(-4\) and \(-102\) to get \(-106\). This is the remainder.

Step 6: Write the Quotient

The quotient is based on the results of synthetic division. The solution row \([3, -11, 34]\) represents \(3x^2 - 11x + 34\). Therefore, the quotient is \(3x^2 - 11x + 34\) with a remainder of \(-106\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial, resulting in a quotient and sometimes a remainder. In our example, we're dividing the polynomial \(3x^3 - 2x^2 + x - 4\) by the binomial \(x + 3\). The process of division follows a set of steps similar to long division but tailored for polynomials.

In synthetic division, a shorter way to perform this division, we focus primarily on the coefficients of the polynomial terms. Given the polynomial \(3x^3 - 2x^2 + x - 4\), the coefficients are \([3, -2, 1, -4]\). This technique simplifies calculations and is especially handy when the divisor is a linear polynomial like \(x + 3\).

• Identify the coefficients from the polynomial.
• Change the sign of the constant (in \(x + 3\), use \(-3\)).
• Proceed with synthetic division by following a specific procedure of multiplication and addition, as demonstrated in our solved exercise.

This technique of polynomial division results in both the quotient and the remainder.

Remainder Theorem

The remainder theorem is a fundamental concept in algebra that relates polynomial division to evaluation at a given point. According to the theorem, when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder \(r\) is equal to \(f(c)\). In the exercise, after we applied synthetic division to \(3x^3 - 2x^2 + x - 4\) with \(x + 3\) as the divisor, we obtained a remainder of \(-106\).

To verify this using the remainder theorem:

• Evaluate the polynomial at \(-3\), i.e., \(f(-3)\).
• Compute \(3(-3)^3 - 2(-3)^2 + (-3) - 4\).
• The result should be \(-106\), confirming the remainder found via synthetic division.

This theorem helps ensure our division is correct, providing a quick way to check remainders without repeating the entire division process.

Quotient

The quotient in polynomial division represents the polynomial you receive after dividing the terms of the dividend by the divisor without considering the remainder. In our example, synthetic division yielded the last row of results, \([3, -11, 34]\).

These numbers become the coefficients of the quotient polynomial. Hence, the quotient derived from the division of \(3x^3 - 2x^2 + x - 4\) by \(x + 3\) is \(3x^2 - 11x + 34\).

• Always arrange terms in decreasing order of the variable's power.
• The quotient provides insights into how the polynomial divides except for the remainder.
• Understanding the quotient helps in simplifying polynomial expressions or solving equations where division is needed.

Note that even if there's a remainder, the quotient is still a crucial outcome of the division process.

Algebra Concepts

Algebra concepts dealing with polynomials are the building blocks of synthetic division and other polynomial operations. It's essential to grasp these concepts fully to understand synthetic division.

Key algebra concepts include:

• Terms and Coefficients: Polynomials are made of terms, each with a coefficient and exponent.
• Degree of a Polynomial: The highest exponent in the polynomial; in \(3x^3 - 2x^2 + x - 4\), the degree is 3.
• Operations with Polynomials: Includes addition, subtraction, multiplication, and division.
• Roots and Zeros: Solutions to equations like \(f(x) = 0\); related closely to the remainder theorem and division of polynomials.

Grasping these concepts helps make synthetic division easier and ensures accurate calculations when working with polynomial equations. Understanding these foundations allows students to navigate more complex algebraic expressions and operations.