Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to find \(P(k).\) $$k=3 ; \quad P(x)=x^{2}-x+3$$

Step-by-Step Solution

Answer

The value of \( P(3) \) is 9.

Step 1: Write down the coefficients

Identify the coefficients of the polynomial \( P(x) = x^2 - x + 3 \). The polynomial is of the form \( ax^2 + bx + c \), so the coefficients are \( a = 1 \), \( b = -1 \), and \( c = 3 \).

Step 2: Set up the synthetic division

For synthetic substitution, we use a table where the top row consists of the coefficients of the polynomial: \( [1, -1, 3] \). We will substitute \( k = 3 \) along the left side of the table.

Step 3: Bring down the first coefficient

Immediately bring down the first coefficient (which is 1) in the synthetic division table. This acts as the starting total.

Step 4: Multiply and add iteratively

Multiply the value we just brought down (1) by \( k = 3 \) and add to the next coefficient (-1): \( 1 \times 3 = 3 \). Adding gives \( -1 + 3 = 2 \). Write 2 as the new second result.

Step 5: Continue multiplication and addition

Now take the result (2) and multiply by \( k = 3 \). \( 2 \times 3 = 6 \). Add this to the final coefficient (3): \( 3 + 6 = 9 \). Write 9 as the final result of this synthetic division process.

Step 6: Interpret the final value

The last number from the synthetic division process is the value of \( P(k) \). Therefore, \( P(3) = 9 \).

Key Concepts

Understanding Polynomial Evaluation

Polynomial evaluation is a core concept in precalculus mathematics. It involves finding the value of a polynomial at a given number, also known as substituting this number into the polynomial. For example, with the polynomial \( P(x) = x^2 - x + 3 \) and given \( k = 3 \), polynomial evaluation requires determining \( P(3) \). This is done by replacing every occurrence of \( x \) with \( 3 \), leading to \( 3^2 - 3 + 3 \).

Calculating step by step:

• Solve \( 3^2 = 9 \)
• Then \( 9 - 3 = 6 \)
• Finally, \( 6 + 3 = 9 \)

So, \( P(3) = 9 \). Evaluating polynomials in this manner often supports deeper learning by illustrating how the polynomial behaves as \( x \) changes.

The Role of Synthetic Division in Polynomial Evaluation

Synthetic division is a technique closely linked to polynomial evaluation, particularly when using synthetic substitution. It provides a streamlined way to evaluate polynomials at given points, leveraging a tabular format to simplify calculations. The process involves:

• Writing the coefficients of the polynomial in a row
• Using the variable \( k \) as the divisor
• Perform interim calculations by multiplying and adding coefficients

This technique cuts down on computational errors by collapsing several operations into a concise format.

In the synthetic substitution with \( P(x) = x^2 - x + 3 \), the coefficients \( [1, -1, 3] \) are used with \( k = 3 \). The calculations inside the synthetic division table directly yield the value of the polynomial, avoiding the traditional multiple substitution and manual arithmetic operations. Ultimately, the last number computed, \( 9 \), represents \( P(3) \).

Synthetic Substitution in Precalculus Mathematics

Precalculus mathematics forms the foundation for advanced algebraic operations, including polynomial evaluation through synthetic substitution. It equips students with efficient methods for complex polynomial calculations which are pivotal in calculus and beyond.

Synthetic substitution, derived from synthetic division, simplifies the evaluation of polynomials at given points. This method reveals relationships and patterns within polynomial functions more clearly.

• This is particularly useful when dealing with higher degree polynomials.
• It supports a deeper understanding of how coefficients affect a polynomial's graph.
• Offers a practical approach to checking polynomial roots efficiently.

As students master synthetic substitution, they build both confidence and proficiency in mathematics, facilitating smoother transitions to more advanced topics and applications. Through consistent practice, this method becomes an invaluable tool in a student's mathematical toolkit.