Problem 6 - Algebra for College Students

Question

For Problems 1-10, find \(f(c)\) by (a) evaluating \(f(c)\) directly, and (b) using synthetic division and the remainder theorem. $$ f(x)=3 x^{4}-4 x^{3}+5 x^{2}-7 x+6 \text { and } c=1 $$

Step-by-Step Solution

Answer

\( f(1) = 3 \).

Step 1: Evaluate \( f(c) \) Directly

To find \( f(c) \) directly, we substitute \( c = 1 \) into the function \( f(x) = 3x^4 - 4x^3 + 5x^2 - 7x + 6 \):\[ f(1) = 3(1)^4 - 4(1)^3 + 5(1)^2 - 7(1) + 6 \]This simplifies to:\[ f(1) = 3(1) - 4(1) + 5(1) - 7(1) + 6 \]\[ f(1) = 3 - 4 + 5 - 7 + 6 \]\[ f(1) = 3 \]So, \( f(1) = 3 \).

Step 2: Set up Synthetic Division

To use synthetic division, prepare the coefficients of the polynomial \( f(x) = 3x^4 - 4x^3 + 5x^2 - 7x + 6 \). The coefficients are 3, -4, 5, -7, and 6. Write them in a row:3 \(-4\) 5 \(-7\) 6Since we are evaluating at \( c = 1 \), write \( 1 \) to the left to indicate division by \( (x - 1) \).

Step 3: Perform Synthetic Division

1. Bring down the leading coefficient (3) unchanged.2. Multiply the number you just brought down by \( c = 1 \), and write the result below the next coefficient:theory sequence -43 12 17\[3 \quad -4 \quad 5 \quad -7 \quad 6 \]\[ \ \quad \quad \quad \quad \quad \quad \]3. Add, write the result below:4. Repeat the process:1 × 3 = 3 => -4 + 3 = -12. Bring down the next coefficient (5), add:1 × -1 = -1 => 5 + -1 = 43. Continue with the next coefficient (-7):1 × 4 = 4 => -7 + 4 = -34. Lastly, continue the process with the last coefficient (6):1 × -3 = -3 => 6 + -3 = 3The remainder is 3.

Step 4: Conclude with Remainder Theorem

According to the remainder theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). Therefore, using the result from the synthetic division, \( f(1) = 3 \).Both methods confirmed: \( f(1) = 3 \).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of long division used in polynomials. It allows you to divide a polynomial by a linear factor of the form \((x - c)\) efficiently. This method is preferred as it requires less writing and fewer calculations compared to traditional long division.
To perform synthetic division:

• Write down the coefficients of the polynomial in descending order of the exponent.
• Place the value of \(c\), from \((x - c)\), on the left side. This will be our divider.
• Bring down the leading coefficient to the lower row.
• Multiply the leading coefficient by \(c\), place the result under the next coefficient, and add them.
• Continue this process for each coefficient, finally reaching the remainder, which confirms \(f(c)\).

This method is handy as it also confirms solutions provided by other methods like polynomial evaluation directly. It isn't just faster but also supports understanding topics like the Remainder Theorem.

Polynomial Evaluation

Evaluating a polynomial directly means substituting a given value into the polynomial equation and computing the result. In the context of the problem, we're given a polynomial \(f(x) = 3x^4 - 4x^3 + 5x^2 - 7x + 6\) and evaluated it at \(c = 1\).
The process is straightforward:

• Substitute \(c\) into each term of the polynomial.
• Calculate the value of each term with the substitution.
• Finally, sum up all the terms to get \(f(c)\).

This method can sometimes feel tedious, especially with higher degree polynomials, but it's a reliable way to find the value of the polynomial at a specific point. This approach also helps verify results obtained from other polynomial methods like synthetic division.

Coefficients of Polynomial

The coefficients in a polynomial play a vital role in expressing how different terms contribute to the overall function. In our polynomial \(f(x) = 3x^4 - 4x^3 + 5x^2 - 7x + 6\):

• The coefficients are 3, -4, 5, -7, and 6, aligned with the powers of \(x\) from highest to lowest.
• In synthetic division, these coefficients are used to set up the division process. They appear in a row indicating their respective contribution to the polynomial function.
• By manipulating these coefficients through division processes like synthetic division, we can determine the behavior of the polynomial efficiently.

Understanding coefficients is crucial as they determine the polynomial's shape and roots, and they are used extensively in computational methods across mathematical studies.

Division of Polynomial

The division of polynomials involves finding how many times one polynomial (the divisor) fits into another polynomial (the dividend). The goal is often to identify quotients and remainders.
When employing division techniques:

• Traditional long division treats polynomials similarly to numerical division, aligning terms and subtracting down.
• Synthetic division streamlines the process by focusing on coefficients and significantly reducing complexity.
• In both methods, understanding how to handle remainders is critical, as the Remainder Theorem ties these results to polynomial evaluation.

Mastering polynomial division techniques is advantageous for solving equations, analyzing polynomial functions, and simplifying complex mathematical expressions.