Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(3 x^{3}-2 x^{2}+x-4\right) \div(x+3) $$

Step-by-Step Solution

Answer

The remainder is \(-106\).

Step 1: Identify the divisor and coefficient substitution

The Remainder Theorem states that the remainder of a polynomial \( f(x) \) divided by \( x-a \) is \( f(a) \). First, identify the divisor \( x+3 \) and rewrite it in the form \( x-a \). Here, \( x-a = x-(-3) \), so \( a = -3 \).

Step 2: Substitute \( a \) into the polynomial

With \( a = -3 \), substitute \(-3\) into the polynomial \( 3x^3 - 2x^2 + x - 4 \). This becomes \( 3(-3)^3 - 2(-3)^2 + (-3) - 4 \).

Step 3: Calculate each term of the polynomial

Calculate each term step by step: - \( 3(-3)^3 = 3(-27) = -81 \)- \( -2(-3)^2 = -2(9) = -18 \)- \( (-3) = -3 \)- \( -4 = -4 \)

Step 4: Sum the results

Add the results from Step 3 to find the remainder: - Sum: \( -81 - 18 - 3 - 4 = -106 \)

Step 5: Conclude with the remainder result

The remainder when \( 3x^3 - 2x^2 + x - 4 \) is divided by \( x+3 \) is \(-106\).

Key Concepts

Polynomial Division

Polynomial division is very similar to regular long division, but instead of numbers, we work with polynomials. In polynomial division, we have a dividend (the polynomial being divided) and a divisor (the polynomial you are dividing by). The process can be used to simplify polynomials, find remainders, or even solve polynomial equations.

• Dividing a polynomial by a monomial is straightforward and involves dividing each term of the polynomial individually.
• When dividing by another polynomial, the process becomes more complex, often involving steps similar to long division known from numbers.

It is important to carefully apply this process and follow each step to avoid calculation errors. Polynomial division can be lengthy, but it provides insights into the interaction between different polynomials. Plus, it lays the groundwork for understanding more advanced algebraic concepts.

Divisors

In the context of polynomials, a divisor is the polynomial by which you are dividing another polynomial. Understanding divisors is crucial for executing polynomial division and utilizing the Remainder Theorem effectively.

• For example, in the polynomial division of \(3x^3 - 2x^2 + x - 4\) by \(x+3\), \(x+3\) acts as the divisor.
• Divisors help determine the quotient (result of division) and remainder when dividing polynomials.

A key point about divisors, in the context of the Remainder Theorem, is being able to express them in the form \(x - a\). This helps in easily calculating the remainder via polynomial substitution, as demonstrated in the exercise.

Polynomial Substitution

Polynomial substitution is a method where you replace the variable in a polynomial with a specific number. This is used to evaluate the polynomial at that particular value of the variable. It plays a vital role in the Remainder Theorem.

• For instance, if you have a polynomial \(f(x)\) and want to find the value at \(x = -3\), you substitute \(-3\) wherever \(x\) appears in the polynomial.
• The substitution helps in applying the Remainder Theorem, as it approximates the division result without actually performing long division.

This method simplifies calculations and is particularly useful when dealing with high degree polynomials, as it allows for direct computation of the remainder.

Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically useful when dividing by a linear factor such as \(x - a\). It streamlines the process by reducing the amount of work needed to divide polynomials.

• It involves using coefficients of the polynomial, carrying out simple arithmetic operations, and works efficiently for polynomials with missing terms.
• Synthetic division especially shines when utilizing the Remainder Theorem as it provides a quick route to determining the remainder.

In our example problem, however, we utilized direct substitution and calculation instead of synthetic division. Nevertheless, being familiar with synthetic division can save time and effort in many polynomial division scenarios.