Problem 343 - Precalculus

Question

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$x-\frac{1}{2}, 2 x^{4}-x^{3}+2 x-1$$

Step-by-Step Solution

Answer

Yes, \( x - \frac{1}{2} \) is a factor, factorization is \((x - \frac{1}{2})(2x^3 + 2)\).

Step 1: Set up for Synthetic Division

Identify the root from the first expression. The divisor is \( x - \frac{1}{2} \), which means the root is \( \frac{1}{2} \). We will use this value for the synthetic division. The dividend is given as the polynomial \( 2x^4 - x^3 + 0x^2 + 2x - 1 \). Note the placeholder for the missing \( x^2 \) term, which is \( 0x^2 \).

Step 2: Perform Synthetic Division

Write the coefficients of the polynomial: \( 2, -1, 0, 2, -1 \). Start the synthetic division by bringing down the leading coefficient, which is 2. Then, multiply \( \frac{1}{2} \) by the value just brought down and continue the process with all coefficients:- Bring down \( 2 \).- Multiply \( \frac{1}{2} \cdot 2 = 1 \); add to \(-1\), giving \( 0 \).- Multiply \( \frac{1}{2} \cdot 0 = 0 \); add to \( 0 \), giving \( 0 \).- Multiply \( \frac{1}{2} \cdot 0 = 0 \); add to \( 2 \), giving \( 2 \).- Multiply \( \frac{1}{2} \cdot 2 = 1 \); add to \(-1\), giving \( 0 \).Thus, the remainder is 0.

Step 3: Conclude Based on Remainder

Since the remainder is 0, \( x - \frac{1}{2} \) is a factor of \( 2x^4 - x^3 + 2x - 1 \). The results of the synthetic division gives us the coefficients \( 2, 0, 0, 2 \), which form the quotient polynomial \( 2x^3 + 2 \). Therefore, the factorization of the original polynomial is:\[ (x - \frac{1}{2})(2x^3 + 2) \]

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler polynomials (called factors) that, when multiplied together, give the original polynomial. This is similar to prime factorization with numbers. By factoring a polynomial, we can understand its properties better, such as its behavior and roots.

In the context of our problem, by using synthetic division, we determined that \(x - \frac{1}{2}\) is indeed a factor of \(2x^4 - x^3 + 2x - 1\). The result of the factorization was given by

• The factor \(x - \frac{1}{2}\)
• The quotient polynomial \(2x^3 + 2\)

Therefore, the factorization is \((x - \frac{1}{2})(2x^3 + 2)\).

This process helps in simplifying complex polynomial expressions and is fundamental in solving polynomial equations.

Roots of Polynomials

The roots of a polynomial are the values of \(x\) for which the polynomial equals zero. These are closely tied to the factors of the polynomial.

For example, if a polynomial has a factor \(x - a\), then \(a\) is a root of the polynomial. In our solution, since \(x - \frac{1}{2}\) is a factor, \(\frac{1}{2}\) is a root of the polynomial \(2x^4 - x^3 + 2x - 1\).

Finding the roots is essential, as they give insights into the graph of the polynomial, helping to identify where it crosses or touches the x-axis. It is an important concept in algebra and calculus, vital for solving equations and understanding functions.

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, producing a quotient and possibly a remainder. It is analogous to numerical division.

There are two main methods:

• Long division: a detailed method similar to long division with numbers.
• Synthetic division: a simplified form focusing on coefficients, great for divisors of the form \(x - a\).

In our exercise, synthetic division was used because it is efficient when dealing with linear divisors. We aligned the coefficients of the dividend polynomial \(2, -1, 0, 2, -1\) and iterated through them using the root \(\frac{1}{2}\) as our divisor.

The end results were simpler polynomials that allowed us to confirm \(x - \frac{1}{2}\) as a factor, which streamlined the factorization process. Polynomial division is crucial for simplifying expressions and solving polynomial equations.