Problem 36 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\(P(2) = -2\)

Step 1: Set up the synthetic division table

Write down the coefficients of the polynomial \(P(x) = 2x^3 - 3x^2 - 5x + 4\), which are \(2, -3, -5, 4\). Place the value of \(k = 2\) to the left of the coefficients.

Step 2: Bring down the leading coefficient

Bring down the first coefficient, 2, as it is. This is the start of our synthetic division process. The "brought down" number acts as the first partial quotient.

Step 3: Multiply and add

Multiply \(2\) (the value of \(k\)) by \(2\) (the number just brought down) to get \(4\), and write it under the next coefficient, \(-3\). Then, add \(-3 + 4 = 1\), and write \(1\) below the line.

Step 4: Repeat multiplication and addition

Take this new sum \(1\) and multiply by \(k = 2\) to get \(2\). Write it under the following coefficient, \(-5\). Add \(-5 + 2 = -3\), and put \(-3\) below the line.

Step 5: Continue the process

Take \(-3\) (the latest sum) and multiply by \(2\) to get \(-6\). Write it under the last coefficient, \(4\). Add \(4 + (-6) = -2\). This final sum, \(-2\), is \(P(2)\).

Key Concepts

Polynomial Division

Polynomial division is an essential technique in algebra that allows us to divide one polynomial by another, usually a linear polynomial. Two common methods to perform polynomial division are long division and synthetic division. Long division involves a process similar to numeric long division, but synthetic division is a shorthand method that is easier to execute, especially when dividing by a linear binomial.
In synthetic division, the key is to align and manipulate the coefficients of the polynomial. This method is most convenient when dividing by a divisor of form \(x - k\). Synthetic division uses the constants or "roots" to simplify calculations.

• It begins with writing down the coefficients of the polynomial in descending order of power.
• The divisor root is written separately and used to process the coefficients sequentially.
• The process involves multiplying and adding, which reduces potential errors compared to long division.

Synthetic division not only simplifies the division process but also reveals important properties of the polynomial, like roots, without diving into detailed calculation.

Remainder Theorem

The Remainder Theorem is a handy tool in understanding polynomial division and roots. It states that if we divide a polynomial \(P(x)\) by \(x - k\), the remainder of that division is the value of the polynomial evaluated at \(k\). This connects division of polynomials to evaluating them at specific points.
For example, when using synthetic division on \(P(x) = 2x^3 - 3x^2 - 5x + 4\) with \(k = 2\), the remainder is not just any number—it's exactly \(P(2)\). This means that performing the division gives us direct insight into the behavior of the polynomial at \(x = 2\).

• To find this remainder using synthetic division, focus on the final number obtained after processing all coefficients.
• If the remainder is zero, \(x = k\) is a root of the polynomial—the polynomial is divisible by \(x - k\).
• If not, this remainder gives the value of the polynomial when \(x = k\), thus confirming the theorem's assertion.

The Remainder Theorem thus provides a shortcut to evaluate polynomials and understand their divisibility in terms of roots and factors.

Polynomial Evaluation

Polynomial evaluation refers to determining the value of a polynomial function at a certain point. This is a common need in algebra, often simplified by performing synthetic division as seen in problems like finding \(P(k)\).
When evaluating polynomials directly, substitutions can become cumbersome, especially with higher degree polynomials. Synthetic division streamlines this by reducing steps to multiplication and addition, which are more straightforward.

• Use the Remainder Theorem to reduce the need for direct calculation by synthetic division.
• Evaluate higher-degree polynomials quickly by leveraging known properties and theorems.
• Through synthetic division, compute values efficiently while also inspecting divisibility and factorization.

By mastering these methods, you can both evaluate polynomials quickly and extract additional insights about their roots and behavior at specific points.