Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(4 x^{4}-2 x^{3}-4 x+2\right) \div(2 x-1) $$

Step-by-Step Solution

Answer

The quotient is \( 4x^3 - 4 \).

Step 1: Adjust the divisor

The given divisor is \( 2x - 1 \). We need the linear term coefficient to be 1 for synthetic division. Divide both the dividend and the divisor by 2, turning the divisor into \( x - \frac{1}{2} \).

Step 2: Arrange the dividend in standard form

The dividend is \( 4x^4 - 2x^3 + 0x^2 - 4x + 2 \). It's already in standard polynomial form. Note the coefficient for each term: \( 4, -2, 0, -4, 2 \).

Step 3: Set up synthetic division

Use \( \frac{1}{2} \) from the adjusted divisor \( x - \frac{1}{2} \). Write the coefficients from the dividend: \( 4, -2, 0, -4, 2 \). Set up the synthetic division table.

Step 4: Perform synthetic division

- Bring down the 4.- Multiply \( 4 \) by \( \frac{1}{2} \) and write the result under the next coefficient. Add: \( -2 + 2 = 0 \).- Repeat this process with the resulting numbers: - Multiply \( 0 \) by \( \frac{1}{2} \) and write it under the next coefficient (0) and add: \( 0 + 0 = 0 \). - Multiply \( 0 \) by \( \frac{1}{2} \), write it under next coefficient (-4) and add: \( -4 + 0 = -4 \). - Multiply \( -4 \) by \( \frac{1}{2} \), write under next coefficient (2) and add: \( 2 - 2 = 0 \).

Step 5: Interpret the result

The result of the synthetic division is \( 4, 0, 0, -4 \) with no remainder. Thus, the quotient is \( 4x^3 - 4 \).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide expressions where both the dividend and divisor are polynomials. It is similar to the long division method used in arithmetic but applied to algebraic terms.
In this context, the dividend is the polynomial you want to divide, and the divisor is the polynomial you are dividing by.
The process helps simplify expressions and can be used to find zeros of functions or analyze polynomial behavior.
The division can be executed using two main strategies: polynomial long division and synthetic division.
Each technique has its benefits, with synthetic division being a more streamlined method for specific types of divisors.

Quotient

The quotient in polynomial division is the result obtained after completing the division process. It's similar to the answer you get when dividing numbers.
For example, when dividing the polynomial \( 4x^4 - 2x^3 - 4x + 2 \) by \( 2x - 1 \), the quotient is the simplified polynomial, \( 4x^3 - 4 \), achieved through synthetic division.
The process disregards the remainder if it exists for finding the basic quotient.

• The quotient provides insight into how the dividend is composed in relation to the divisor.
• It indicates the polynomial that can be multiplied by the divisor to yield the dividend.
• Quotients are essential for further polynomial analysis, such as finding intercepts and behavior trends.

Polynomial Long Division

Polynomial long division is a reliable technique for dividing polynomials of any degree. It works much like regular long division with numbers, broken into a series of simpler steps.
The primary steps involve dividing the highest degree terms, multiplying, subtracting, and bringing down the next term, iteratively.
Polynomial long division is advantageous because:

• It can handle any type of polynomial divisor.
• It produces both a quotient and a remainder, offering a complete division perspective.
• It builds a strong foundation for understanding more complex algebraic concepts.

Despite being more detailed than synthetic division, it demonstrates a clear, logical way to solve polynomial divisions, particularly when the divisor is not a linear polynomial.

Division of Polynomials

The division of polynomials encompasses various methods, primarily designed to simplify complex polynomial expressions.
Techniques like synthetic and polynomial long division provide different approaches tailored to the nature of the divisor.
Synthetic division is preferred for its simplicity but is largely limited to situations where the divisor is linear in form, like \( x - a \).
On the other hand, polynomial long division is more versatile and applicable to higher-degree divisors.
It's important to understand:

• Each method reduces the polynomial into simpler parts.
• The choice of division technique may depend on divisor properties.
• The results from these divisions lead to essential insights about polynomial behavior.

Mastery of these divisions is crucial for advanced algebraic studies, enabling efficient and effective solving of polynomial equations.