Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to find \(P(k).\) $$k=2 ; \quad P(x)=x^{2}-5 x+1$$

Step-by-Step Solution

Answer

The value of \(P(2)\) is \(-5\).

Step 1: Set up the equation for synthetic substitution

We need to find \(P(k)\) where \(k = 2\) and \(P(x) = x^2 - 5x + 1\). In synthetic substitution, we substitute \(k\) into the polynomial to find its value. We write down the coefficients of \(P(x)\), which are \(1\) for \(x^2\), \(-5\) for \(x\), and \(1\) for the constant term.

Step 2: Write the coefficients and initial row

We write the coefficients of \(P(x)\) as a row: \(1, -5, 1\). Below these coefficients, we will perform the calculations. Leave space for the multiplication and addition steps below each coefficient.

Step 3: Start with the leading coefficient

Bring down the leading coefficient (\(1\)) to the bottom row unchanged. This is your starting point for the synthetic substitution calculation. So, we write \(1\) below the first column.

Step 4: Multiply and add for each coefficient

Multiply \(2\) by the first number in the bottom row (which is \(1\)), and write the result (\(2\)) below the second coefficient (\(-5\)). Then, perform the addition \((-5) + 2 = -3\) and write \(-3\) in the bottom row.

Step 5: Repeat multiply and add for next coefficient

Take \(2\), multiply it by \(-3\) (the last result in the bottom row), which equals \(-6\). Write \(-6\) below the next coefficient \(1\). Add \(1 + (-6) = -5\) and write \(-5\) in the bottom row.

Step 6: Interpret the result

The final number in the bottom row, \(-5\), is \(P(k)\). This means that \(P(2) = -5\).

Key Concepts

Polynomials

A polynomial is an expression made up of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. Polynomials are fundamental in mathematics and have the general form:\[P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]where each \(a_i\) denotes a coefficient, and \(x\) is the variable. Here, \(n\) is the degree of the polynomial, determined by the highest exponent.In our exercise, we handle a simple quadratic polynomial:\[P(x) = x^2 - 5x + 1\]This polynomial has a degree of 2. The coefficients are 1, -5, and 1 for terms \(x^2\), \(x\), and the constant term respectively. Recognizing polynomials' structures helps us manipulate and evaluate them, using methods like substitution.

Substitution Method

The substitution method is a technique used to evaluate a polynomial at a specific value of the variable. This is especially useful when we want to determine the polynomial's value for given \(x\). In our case, we use a focused form of substitution, called synthetic substitution.Synthetic substitution is a streamlined algorithm that simplifies evaluating polynomials, especially useful for checking the remainder when a polynomial is divided by a linear factor.To utilize the substitution method:

• Identify the polynomial and the value to substitute.
• Extract the coefficients of the polynomial.
• Apply the value to these coefficients through sequential multiplication and addition, simplifying the calculations.

Our example uses \(k = 2\) in the polynomial \(P(x) = x^2 - 5x + 1\), demonstrating how synthetic substitution helps efficiently find \(P(k)\) without extensive arithmetic.

Coefficient Calculation

In synthetic substitution, coefficient calculation is key to efficiently determining the value of a polynomial at a given variable value. Here’s how you approach it:

• Begin with writing the coefficients in a row: in our example, these are 1 (for \(x^2\)), -5 (for \(x\)), and 1 (constant term).
• The process involves bringing down the first coefficient to the bottom row unchanged. This becomes the starting point of our calculation.
• Next, use the given \(k = 2\) to sequentially multiply and add, modifying coefficients as you move across.
  • Multiply each result by \(k\) and add the next top-row coefficient. For instance, after multiplying the first brought-down coefficient by \(k\), add the result to the next top coefficient to continue.
  • This sequence is repeated until all coefficients have been processed, leading to the result in the bottom row capturing \(P(k)\).

For our exercise, the calculated result through this series of operations is -5, indicating that \(P(2) = -5\). Coefficient calculation simplifies and organizes this process, minimizing errors and confusion.