Problem 43 - Precalculus: Functions and Graphs

Question

Find the remainder if the polynomial $$3 x^{100}+5 x^{35}-4 x^{36}+2 x^{17}-6$$ is divided by \(x+1\)

Step-by-Step Solution

Answer

-14

Step 1: Understanding the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). In this case, we need to find the remainder of the polynomial when divided by \( x + 1 \). Which means we need to evaluate the polynomial at \( x = -1 \).

Step 2: Substitute and Evaluate the Polynomial

To evaluate the polynomial \( 3x^{100} + 5x^{35} - 4x^{36} + 2x^{17} - 6 \), substitute \( x = -1 \). This gives us: \[ f(-1) = 3(-1)^{100} + 5(-1)^{35} - 4(-1)^{36} + 2(-1)^{17} - 6 \] Now, simplify each term.

Step 3: Simplify Each Term

1. \((-1)^{100} = 1\) because the exponent is even, so \(3(-1)^{100} = 3 \).2. \((-1)^{35} = -1\) because the exponent is odd, so \(5(-1)^{35} = -5 \).3. \((-1)^{36} = 1\) because the exponent is even, so \(-4(-1)^{36} = -4 \).4. \((-1)^{17} = -1\) because the exponent is odd, so \(2(-1)^{17} = -2 \).5. The constant \(-6\) remains as it is.

Step 4: Combine and Simplify the Results

Now combine the simplified terms: \[ 3 - 5 - 4 - 2 - 6 = -14 \] Thus, the remainder when \( 3x^{100} + 5x^{35} - 4x^{36} + 2x^{17} - 6 \) is divided by \( x + 1 \) is \(-14\).

Key Concepts

Polynomial Division

Polynomial division is a process that involves dividing one polynomial by another, much like long division with numbers. It allows us to determine how many times a divisor polynomial fits into the dividend polynomial and what the remainder is. There are a couple of key elements to remember when performing polynomial division:

• Dividend: The polynomial we are dividing up.
• Divisor: The polynomial we are dividing by.
• Quotient: The result of the division process.
• Remainder: What is left over after division.

When the division is complete, the original polynomial can be expressed as: \[ \text{{Dividend}} = (\text{{Divisor}} \times \text{{Quotient}}) + \text{{Remainder}} \]
If the divisor is a simple binomial like \(x + 1\), the problem is sometimes simplified using the Remainder Theorem, which makes it more straightforward to find the remainder without having to complete the entire division process.

Evaluating Polynomials

Evaluating a polynomial means finding its value for a particular value of \(x\). This is especially useful when applying the Remainder Theorem, as it tells us that the remainder when a polynomial \(f(x)\) is divided by \(x - c\) is \(f(c)\).
To evaluate a polynomial:

• Substitute the given value into the polynomial in place of \(x\).
• Calculate the result by simplifying the expression.

For instance, if we have the polynomial \(3x^{100} + 5x^{35} - 4x^{36} + 2x^{17} - 6\) and we need to evaluate it at \(x = -1\) to apply the Remainder Theorem, we substitute \[x = -1 \] into the polynomial and simplify each term accordingly. Understanding this substitution and simplification process is crucial for efficiently finding remainders.

Remainder in Polynomial Division

The Remainder Theorem gives us a powerful tool for quickly finding the remainder when a polynomial is divided by a binomial of the form \(x - c\). It tells us that instead of going through the complete division process, we can simply substitute \(c\) into the polynomial to get the remainder directly.
In our exercise, we used this theorem on the polynomial \(3x^{100} + 5x^{35} - 4x^{36} + 2x^{17} - 6\) divided by \(x + 1\). Since the divisor is \(x + 1\), this fits the form \(x - (-1)\).
Using the theorem:

• Evaluate the polynomial at \(x = -1\).
• The result of this evaluation is the remainder when dividing by \(x + 1\).

For our polynomial, substituting \(x = -1\) yielded a remainder of \(-14\).
This approach simplifies and speeds up the problem-solving process, helping to focus on simplification rather than lengthy division calculations.