Problem 39 - Precalculus with Calculus Previews

Question

Use synthetic division and the Remainder Theorem to find \(f(c)\) for the given value of c. $$ f(x)=2 x^{6}-3 x^{5}+x^{4}-2 x+1 ; c=4 $$

Step-by-Step Solution

Answer

Using synthetic division, \(f(4) = 5369\).

Step 1: Write Down Polynomial and Value of c

The polynomial given is \(f(x) = 2x^6 - 3x^5 + x^4 - 0x^3 - 0x^2 - 2x + 1\). The task is to find \(f(4)\) using synthetic division.

Step 2: Set Up Synthetic Division Table

Create a synthetic division table by placing the coefficients \([2, -3, 1, 0, 0, -2, 1]\) in the top row. Leave a column to the left blank and place the value \(c=4\) to the left.

Step 3: Begin Synthetic Division Process

Bring down the first coefficient, \(2\), as-is. Multiply it by \(4\) (the value of \(c\)), and write the result, \(8\), under the second coefficient, \(-3\).

Step 4: Continue Synthetic Division Calculations

Add \(-3\) and \(8\) to get \(5\). Multiply \(5\) by \(4\) to get \(20\). Write \(20\) under the next coefficient, \(1\).

Step 5: Repeat Addition and Multiplication

Add \(1\) and \(20\) to get \(21\). Multiply \(21\) by \(4\) to get \(84\). Write \(84\) under the next coefficient, \(0\).

Step 6: Continue Process Until Last Coefficient

Add \(0\) and \(84\) to get \(84\). Multiply \(84\) by \(4\) to get \(336\), and write it under the next \(0\). Then add \(0\) and \(336\), multiply by \(4\) to get \(1344\), and put it under the next coefficient, \(-2\).

Step 7: Complete Last Division Operation

Add \(-2\) and \(1344\) to get \(1342\). Multiply by \(4\) to yield \(5368\), putting the final result under the last coefficient, \(1\).

Step 8: Add Final Coefficients for Remainder

Add \(1\) and \(5368\) to obtain \(5369\), which is the remainder and equals \(f(4)\).

Key Concepts

Remainder Theorem

The Remainder Theorem is an incredibly useful tool in algebra that connects polynomial functions and their values to synthetic division. It states that if you divide a polynomial \(f(x)\) by \(x-c\), the remainder of this division is \(f(c)\). This means that instead of solving the entire polynomial function, you can simply use synthetic division to find the remainder, which directly gives you the value of \(f(c)\).
One of the powerful aspects of the Remainder Theorem is its simplicity in evaluating polynomials at given points without having to go through cumbersome calculations. This theorem not only speeds up calculations but also helps in verifying whether a number \(c\) is a root of the polynomial (if \(f(c) = 0\), then \(c\) is a root). It's essentially a shortcut that makes working with polynomial functions much easier and more efficient.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients, structured into terms where the variables are raised to whole number powers. They can take various forms, like the given function \(f(x) = 2x^6 - 3x^5 + x^4 - 2x + 1\).
Each term of a polynomial has a coefficient and a power, and the highest power is known as the degree of the polynomial. In this case, the degree is \(6\), indicating that \(2x^6\) is the leading term.
Understanding polynomial functions is key because they form the basis for algebraic equations and are used in calculus and other advanced mathematical fields. They exhibit varied behaviors depending on their degrees, including how they graph, their end behavior, and how they can be factored or simplified. Recognizing the structure and properties of polynomial functions is crucial for tackling more complex mathematical problems.

Synthetic Division Process

The Synthetic Division Process is a streamlined method to manually divide polynomials, particularly helpful when the divisor is in the form \(x-c\). This process is especially popular due to its simplicity and efficiency compared to traditional long division.
To start, you list out only the coefficients of the polynomial's terms. In our original problem, the coefficients are \([2, -3, 1, 0, 0, -2, 1]\). It's important to include zeros for any missing degrees of \(x\) to maintain consistency.

• First, bring down the leading coefficient directly to start the result row.
• Then, multiply it by \(c\) (here, \(c=4\)) and add it to the next coefficient.
• Continue this process, multiplying and adding across the row until you finish with the last coefficient.

The last number you get after completing these operations provides the remainder of the division, which is also \(f(c)\). This remainder, here calculated as \(5369\), tells us exactly what \(f(4)\) equals, showing how effective synthetic division can be in evaluating polynomial functions.