Problem 386 - Precalculus

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(3 x^{3}+4 x^{2}-8 x+2\right) \div(x-3) $$

Step-by-Step Solution

Answer

The remainder is 95.

Step 1: Identify the Dividend and the Divisor

In the polynomial division problem, the dividend is the polynomial we want to divide, which is \(3x^3 + 4x^2 - 8x + 2\). The divisor is \(x - 3\).

Step 2: Apply the Remainder Theorem

The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder of the division is \(f(c)\). Here, the divisor is \(x - 3\), so \(c = 3\).

Step 3: Substitute c into the Polynomial

Substitute \(x = 3\) into the polynomial: \(3(3)^3 + 4(3)^2 - 8(3) + 2\).

Step 4: Calculate the Expression

Now calculate the expression:- \(3(3)^3 = 81\)- \(4(3)^2 = 36\)- \(-8(3) = -24\)- Combine these with \(+2\):\(81 + 36 - 24 + 2 = 95\).

Step 5: Determine the Remainder

The calculated value of the expression (\(95\)) is the remainder when \(3x^3 + 4x^2 - 8x + 2\) is divided by \(x - 3\).

Key Concepts

Polynomial Division

In mathematics, polynomial division is similar to dividing numbers. It involves dividing one polynomial, known as the dividend, by another polynomial, called the divisor. Importantly, this process is used to determine how many times the divisor fits into the dividend and what is left over. This leftover is called the remainder. Polynomial division can often seem complex due to the various operations involved, but by following systematic steps, it becomes a manageable process. To solve a polynomial division problem, one typically uses either long division or synthetic division. Both methods help break down the problem into smaller, more manageable steps.

Dividend and Divisor

The terms dividend and divisor are key to understanding polynomial division. The dividend is the polynomial you are dividing. In our example, the dividend is the polynomial \(3x^3 + 4x^2 - 8x + 2\). This is the polynomial you want to separate into parts based on the divisor's influence.
The divisor, on the other hand, is the polynomial you are dividing by. In this case, the divisor is \(x - 3\). By understanding the roles of the dividend and divisor, you can effectively apply the Remainder Theorem and more easily solve polynomial division problems, leading you to the remainder quickly.

Substitution in Polynomials

Substitution in polynomials is a method where you replace the variable in the polynomial with a specific value. This concept is crucial when applying the Remainder Theorem. You simply substitute the value of \(c\) from the divisor \(x - c\) into the original polynomial.
In our problem, with the divisor \(x - 3\), the value of \(c\) is 3. You substitute \(x = 3\) into the polynomial \(3x^3 + 4x^2 - 8x + 2\) to simplify and calculate the remainder. Substitution helps transform a complex polynomial into a simpler numerical expression, making it easier to evaluate.

Calculation of Remainder

To find the remainder using the Remainder Theorem, you perform a straightforward calculation. After substituting the value of \(x\) into the polynomial, you will have a numerical expression to solve.
For our polynomial, the calculation proceeds as follows:

• Calculate \(3(3)^3 = 81\)
• Find \(4(3)^2 = 36\)
• Compute \(-8(3) = -24\)

Finally, sum these results along with the constant term 2, yielding \(81 + 36 - 24 + 2 = 95\). Thus, the remainder of dividing \(3x^3 + 4x^2 - 8x + 2\) by \(x - 3\) is 95. This calculation highlights how every component in the expression contributes to the result.