Problem 37 - Precalculus with Calculus Previews

Question

Use synthetic division and the Remainder Theorem to find \(f(c)\) for the given value of c. $$ f(x)=14 x^{4}-60 x^{3}+49 x^{2}-21 x+19 ; c=1 $$

Step-by-Step Solution

Answer

\(f(1) = 1\)

Step 1: Set Up Synthetic Division

Write down the coefficients of the polynomial \(f(x) = 14x^4 - 60x^3 + 49x^2 - 21x + 19\). These coefficients are: 14, -60, 49, -21, and 19. We will use the value \(c = 1\) in synthetic division.

Step 2: Perform Synthetic Division

Write the value \(c = 1\) on the left side of a vertical bar. Next to it, write the coefficients of the polynomial: 14, -60, 49, -21, and 19. Begin the synthetic division process by bringing down the first coefficient, 14.1. Bring down the first coefficient: 14.2. Multiply 14 by 1, (the value of \(c\)), and write 14 under the second coefficient, -60.3. Add -60 and 14 to get -46. Write -46 underneath.4. Multiply -46 by 1, and write -46 under the third coefficient, 49.5. Add 49 and -46 to get 3. Write 3 underneath.6. Multiply 3 by 1, and write 3 under the fourth coefficient, -21.7. Add -21 and 3 to get -18. Write -18 underneath.8. Multiply -18 by 1, and write -18 under the last coefficient, 19.9. Add 19 and -18 to get 1. Write 1 underneath.The sequence of numbers now reads: 14, -46, 3, -18, 1.

Step 3: Interpret the Remainder

The final number in the synthetic division process, 1, represents the remainder. According to the Remainder Theorem, \(f(c) = f(1) = 1\).

Key Concepts

Polynomial Function

A polynomial function is an expression made up of variables and coefficients, structured by addition, subtraction, multiplication, and non-negative integer exponents. These functions are used to represent a wide variety of situations and processes. For instance, the polynomial function in our exercise is \( f(x) = 14x^4 - 60x^3 + 49x^2 - 21x + 19 \). Here are a few things to notice about polynomial functions:

• Each term in the polynomial has a coefficient and a variable raised to an exponent.
• The degree of the polynomial is determined by the largest exponent; in our example, this is 4.
• Polynomials are classified by this degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), quartic (degree 4), and so on.

Understanding the structure and rules of polynomial functions is crucial, as it sets the stage for applying methods such as synthetic division and the Remainder Theorem.

Remainder Theorem

The Remainder Theorem provides a quick way of evaluating polynomials by connecting division and evaluation. It states that when a polynomial \( f(x) \) is divided by \( x-c \), the remainder of this division is exactly \( f(c) \). So, for the given polynomial \( f(x) = 14x^4 - 60x^3 + 49x^2 - 21x + 19 \), to find \( f(1) \), we can use synthetic division.This theorem is particularly useful because it allows us to assess polynomial values without extensive computation. It affirms that evaluating the polynomial at \( c \) gives us the remainder when divided by \( x-c \). This insight is what simplifies polynomial evaluations and is a key step in proving polynomial identities and solving equations.Keep in mind:

• This method allows for verification of roots or zeroes since if \( f(c) = 0 \), then \( x-c \) is a factor of the polynomial.
• It simplifies calculations significantly, especially for high degree polynomials.

Synthetic Division Process

Synthetic division is a simplified, faster way of dividing polynomials, especially useful for dividing by linear factors like \( x-c \). Here, we'll focus on how to use this process effectively and understand its components.The first step in synthetic division is to list the coefficients of the polynomial. For instance, in \( f(x) = 14x^4 - 60x^3 + 49x^2 - 21x + 19 \), the coefficients are 14, -60, 49, -21, and 19. We place these coefficients next to our value of \( c = 1 \) used for division.As we initiate synthetic division:

• First, bring down the initial coefficient, which starts your result line.
• Multiply this number by \( c \) (here, 1) and write it under the next coefficient.
• Add downwards, the sum becomes the next number in your result line, and continue the process.
• Keep multiplying and adding until you reach the last coefficient.

The remainder is the final number in your results, equaling \( f(c) \). This powerful technique breaks division into manageable steps, often used in conjunction with the Remainder Theorem for quick evaluations.