Problem 383 - Precalculus

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(-3 x^{2}+6 x+24\right) \div(x-4) $$

Step-by-Step Solution

Answer

The remainder is 0.

Step 1: Identify Dividend and Divisor

The polynomial to be divided is the dividend, \(-3x^2 + 6x + 24\), and the divisor is \(x - 4\). We'll use the Remainder Theorem here, which states that the remainder of the division of a polynomial \(f(x)\) by \(x - c\) is \(f(c)\). In this case, \(c = 4\).

Step 2: Substitute to Evaluate the Polynomial

According to the Remainder Theorem, substitute \(x = 4\) into the polynomial \(f(x) = -3x^2 + 6x + 24\).

Step 3: Calculate for x=4

Substitute \(x=4\) into the expression: \(-3(4)^2 + 6(4) + 24\).

Step 4: Simplify the Expression

Calculate \(-3(4)^2 = -3 \times 16 = -48\), \(6(4) = 24\), and add 24. The expression becomes \(-48 + 24 + 24\).

Step 5: Calculate the Final Result

Now compute the sum: \(-48 + 24 + 24 = 0\). Thus, the remainder is \(0\).

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers, but instead, we work with expressions. When we perform polynomial division, we divide one polynomial (the dividend) by another (the divisor). This process helps us determine how many times the divisor fits into the dividend and what the remainder is.

• The dividend is the polynomial we want to divide up.
• The divisor is the polynomial we are dividing by.
• This division can result in a quotient (the result of the division) and possibly a remainder (what's left over).

An important aspect of polynomial division is understanding that the operations of addition, subtraction, and multiplication are required to simplify the given expressions. In this context, the Remainder Theorem offers a shortcut for finding the remainder without performing traditional division.

Dividend and Divisor

In the context of polynomial division, recognizing the dividend and divisor forms the first critical step. The dividend is your starting polynomial - In our example, the polynomial \( -3x^2 + 6x + 24 \) is the dividend. - The divisor is the polynomial by which you are dividing. For this division, it is \( x - 4 \). Knowing these terms is essential because, in polynomial operations, different terms possess distinct roles. The dividend undergoes division by the divisor to explore possible factors or remainders, using methods like synthetic division or the Remainder Theorem. By understanding the roles of these components, managing calculations becomes a smoother process.

Substitution

Substitution is a crucial technique in algebra that allows us to replace a variable with a number to simplify the evaluation of expressions. When applying the Remainder Theorem, substitution enables us to calculate the remainder quickly by substituting the root from the divisor into the dividend polynomial.Let's consider the divisor \( x - 4 \).

• Solve \( x - 4 = 0 \) to find \( x = 4 \).
• This value can be substituted back into the dividend polynomial \( f(x) = -3x^2 + 6x + 24 \) to find \( f(4) \).

With this approach, the remaining calculations become straightforward, and they yield the remainder without needing a full division.

Evaluating Polynomials

Evaluating a polynomial involves calculating its value at a particular point. In the process of finding remainders with the Remainder Theorem, evaluating involves substituting the root derived from the divisor into the polynomial to solve for the remainder.
In the given example:

• The root \( x = 4 \) from the divisor \( x - 4 \) is used to evaluate \( f(x) = -3x^2 + 6x + 24 \).
• When substituting \( x = 4 \) into the polynomial, carry out the operations: \[ -3(4)^2 + 6(4) + 24 \].
• Simplify this expression: calculate \( -3 \times 16 = -48 \), calculate \( 6 \times 4 = 24 \), and add 24.
• The ultimate result \( -48 + 24 + 24 = 0 \) gives the remainder.

This result confirms that the divisor \( x - 4 \) is a perfect factor of the dividend polynomial, resulting in no remainder.