Problem 7 - College Algebra

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

In this problem, you need to use the Remainder Theorem to find the remainder when dividing the polynomial \(3x^3 - 2x^2 + x - 4\) by \(x + 3\). According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x - a\), the remainder is \(f(a)\). Here, the divisor is \(x + 3\), which means \(a = -3\) (because \(x + 3\) can be rewritten as \(x - (-3)\)).

Step 2: Substitute the Value into the Polynomial

Now we need to substitute \(-3\) for \(x\) into the polynomial \(3x^3 - 2x^2 + x - 4\). This means calculating \(f(-3)\).

Step 3: Evaluate the Polynomial at \(x = -3\)

Substitute \(x = -3\) into the polynomial: \(f(-3) = 3(-3)^3 - 2(-3)^2 + (-3) - 4\).

Step 4: Calculate Each Term

Calculate the value of each term:1. \(3(-3)^3 = 3(-27) = -81\) 2. \(-2(-3)^2 = -2(9) = -18\) 3. \(-3\)4. \(-4\).

Step 5: Add the Results

Now add all the calculated results together:\(-81 - 18 - 3 - 4 = -106\).

Step 6: Conclusion

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

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division, but it's applied to polynomials instead of integers. The idea is to divide a polynomial by another polynomial of a lesser degree, finding a quotient and a remainder. Think of it as breaking down a more complex problem into simpler parts. The result when you divide one polynomial by another is usually a quotient polynomial and a remainder, which could be a constant or a polynomial of lesser degree than the divisor.

For the exercise, we're interested in the remainder after dividing the polynomial \(3x^3 - 2x^2 + x - 4\) by \(x + 3\). Using polynomial division directly could be time-consuming, but thankfully, the Remainder Theorem offers a shortcut.

Synthetic Substitution

Synthetic substitution is a streamlined method to evaluate polynomials, particularly useful when applying the Remainder Theorem. Instead of substituting values directly into the polynomial and calculating each term separately, synthetic substitution allows quicker calculations.

• Write down the coefficients of the polynomial: For \(3x^3 - 2x^2 + x - 4\), write \([3, -2, 1, -4]\).
• Write the opposite of the constant from the divisor, \(-3\), as you'll substitute it into the polynomial.
• Perform synthetic division, bringing down the first coefficient and use it to systematically calculate the rest.

By using synthetic substitution, you bypass some of the manual processes, making it easier to find the remainder without fully dividing the polynomial.

Polynomial Remainder

The polynomial remainder is the result we obtain when a polynomial is divided by another polynomial. According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x-a\), the remainder is simply \(f(a)\). That means instead of dividing the polynomial completely, we can evaluate the polynomial at \(a\) to find the remainder.

In our exercise, we evaluated \(3x^3 - 2x^2 + x - 4\) at \(-3\) and found the remainder to be \(-106\). Understanding the role of polynomial remainders is crucial as they help in checking divisibility and other polynomial properties.

Divisor Identification

Divisor identification is the key first step when working with polynomial division and the Remainder Theorem. The divisor determines the values you'll work with in subsequent calculations. To use the Remainder Theorem effectively, the divisor needs to be of the form \(x-a\).

• In the given problem, the divisor is \(x+3\).
• To apply the Remainder Theorem, rewrite it as \(x - (-3)\), indicating that \(a = -3\). This step is crucial as it sets the stage for applying the remainder theorem efficiently.

Correct identification ensures accurate substitution and evaluation of the polynomial, making the entire process both efficient and accurate.