Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to find \(P(k).\) $$k=2 ; \quad P(x)=2 x^{3}-3 x^{2}-5 x+4$$

Step-by-Step Solution

Answer

The result of \(P(2)\) is \(-2\).

Step 1: Set Up the Synthetic Division Table

To apply synthetic substitution for finding \(P(k)\), where \(k=2\), set up a table. Write the coefficients of the polynomial \(P(x) = 2x^3 - 3x^2 - 5x + 4\) in a row: 2, -3, -5, 4.

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient, 2, to the bottom row of the table. This is the initial step of synthetic division.

Step 3: Multiply and Add to Continue the Process

Multiply the number you just brought down (2) by \(k\) (which is 2), and write the result under the next coefficient. This result is \(2 \times 2 = 4\). Add this result to the next coefficient, \(-3\), giving \((4) + (-3) = 1\). Write this sum in the row of results.

Step 4: Continue the Process with the New Value

Repeat Step 3 with the new value (1). Multiply 1 by \(2\) to get \(2\). Add \(2\) to the next coefficient, \(-5\), to get \(-3\). Write \(-3\) as the new result.

Step 5: Calculate the Final Result

For the last step, repeat the multiplication with \(-3\). Multiply \(-3\) by \(2\) to get \(-6\). Add \(-6\) to the last coefficient, 4, resulting in \(-2\). Write \(-2\) in the result row.

Key Concepts

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial function when a certain value is substituted for the variable, typically represented as \(x\). In essence, it involves calculating the output of the polynomial by inserting a specific number in place of \(x\). This is essential in understanding how polynomials behave under different inputs and is frequently used to find specific function values.
For instance, if given \(P(x) = 2x^3 - 3x^2 - 5x + 4\) and asked to evaluate \(P(2)\), this means substituting \(2\) in place of \(x\) and performing arithmetic calculations.

The process can be cumbersome with higher degree polynomials or more complex numbers. To simplify this, synthetic substitution provides a more streamlined method, especially useful when systematically exploring the polynomial's behavior at several points.

Synthetic Division

Synthetic division is a shortcut method of dividing a polynomial by a linear divisor of the form \(x - k\). This method is more efficient than traditional long division, particularly with complex polynomials.
Even though the primary usage of synthetic division is to simplify polynomial division, it offers a streamlined technique for polynomial evaluation, known as synthetic substitution. This approach is especially effective when evaluating polynomials at specific points.
Here's how synthetic division works:

• Set-Up: Start by writing the coefficients of the polynomial in descending order of power.
• Operation: Bring down the first coefficient. Then, iteratively multiply this coefficient by \(k\), add it to the next coefficient, and repeat the process.
• Result: The final number in this process gives \(P(k)\), essentially evaluating the polynomial at that point.

Synthetic division simplifies computations, reducing potential errors that could arise from handling multiple terms during calculations.

Polynomials

Polynomials form a fundamental part of algebra and consist of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, a polynomial in \(x\) might look like \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.
Polynomials can vary in complexity:

• Monomials: Single term expressions, like \(3x^2\).
• Binomials: Two-term expressions, such as \(2x + 5\).
• Trinomials: Three-term expressions, for instance, \(x^2 - 3x + 2\).

The degree of the polynomial is determined by the highest power of the variable. This degree is important because it affects the shape and behavior of the polynomial's graph. Understanding polynomials and their properties is crucial for solving algebraic equations and modeling various real-world scenarios.