Problem 9 - Precalculus: Functions and Graphs

Question

Use the remainder theorem to find \(f(c)\). $$f(x)=3 x^{3}-x^{2}+5 x-4 ; \quad c=2$$

Step-by-Step Solution

Answer

The remainder is 26, so \( f(2) = 26 \).

Step 1: Understand the Remainder Theorem

The Remainder Theorem tells us that if a polynomial \( f(x) \) is divided by \( x-c \), then the remainder of this division is \( f(c) \). In simpler terms, \( f(c) \) can be found by substituting \( c \) for \( x \) in \( f(x) \).

Step 2: Substitute c into the polynomial

Substitute \( x = 2 \) into the function \( f(x) = 3x^3 - x^2 + 5x - 4 \). This means we calculate \( f(2) \).

Step 3: Calculate each term in the polynomial

Calculate each term individually:- The first term: \( 3(2)^3 = 3 \times 8 = 24 \)- The second term: \(- (2)^2 = -4 \)- The third term: \( 5 \times 2 = 10 \)- The constant term: \(-4 \)

Step 4: Combine the results

Combine the results from the calculations:- Starting with the first term: \( 24 \)- Add the second term: \( 24 - 4 = 20 \)- Add the third term: \( 20 + 10 = 30 \)- Add the constant term: \( 30 - 4 = 26 \)

Step 5: Conclude with the remainder

Since \( f(2) = 26 \), according to the Remainder Theorem, the remainder when \( f(x) \) is divided by \( x-2 \) is 26.

Key Concepts

Polynomial Division

Polynomial division is similar to long division but applied to polynomials. It involves dividing one polynomial by another, typically in the form of \( x - c \). The division finds out how many times the divisor can go into the dividend (the polynomial you are dividing).
To perform polynomial division, you:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the quotient and subtract it from the dividend.
• Repeat the process with the new polynomial formed until the degree of the remainder polynomial is less than the degree of the divisor.

This process often results in a quotient and a remainder. The Remainder Theorem directly ties into this, telling us that the remainder is actually \( f(c) \) when the polynomial \( f(x) \) is divided by \( x-c \). So instead of performing the whole division, you just have to substitute the value \( c \) into the polynomial.

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial at a specific point. It's a straightforward task of substituting a given value for the variable in a polynomial.
This is how you would evaluate a polynomial:

• Identify the polynomial equation and the point at which to evaluate.
• Substitute the point into the polynomial in place of the variable.
• Perform the mathematical operations to get the result.

For instance, if you need to evaluate a polynomial \( f(x) = 3x^3 - x^2 + 5x - 4 \) at \( x = 2 \), you replace every instance of \( x \) with 2. By calculating each part, you get the final value which, in our example, equals 26. This method closely relates to evaluating whether a number is a root of the polynomial or not.

Synthetic Substitution

Synthetic substitution is a technique that simplifies the process of evaluating polynomials. This method is especially useful when you are dealing with large polynomials or many evaluation points.
Here's how synthetic substitution works:

• Write down the coefficients of the polynomial.
• Use the value of \( c \) as a starting point next to your coefficients.
• Begin the process by bringing down the leading coefficient as it is.
• Multiply the value of \( c \) by the value underneath the line and write this under the next coefficient.
• Add this product to the next coefficient and write the result below the line.
• Repeat the process for each coefficient.

The final result under the last coefficient will represent \( f(c) \), the value of the polynomial at that point. In the example presented, if \( f(x) = 3x^3 - x^2 + 5x - 4 \) and we use \( c = 2 \), performing synthetic substitution will efficiently give us the remainder, which in this case is 26. This confirms what we found using the direct substitution method.