Problem 64 - College Algebra

Question

Synthetic division is a process for dividing a polynomial by \(x-c .\) The coefficient of \(x\) is \(1 .\) How might synthetic division be used if you are dividing by \(2 x-4 ?\)

Step-by-Step Solution

Answer

To use synthetic division with a divisor like \(2x-4\), we first rewrite the divisor in the form \(x-c\), and adjust the polynomial we're dividing by dividing each term by the coefficient of \(x\) in the divisor. Synthetic division is then performed, and the results doubled to give the original quotient and remainder when dividing by \(2x - 4\).

Step 1: Identify divisor equivalence

Since \(2x - 4\) isn't in the form of \(x - c\), we first need to rewrite it in that form. To do so, factor out 2 from \(2x - 4\) to get \(2(x - 2)\). This gives us our \(c\) value, which is \(2\). Now, we can recognize that \(2(x - 2)\) is equivalent to \(2*(x - 2)\). This step is important because it brings the divisor to a form that we can use in synthetic division.

Step 2: Adjust the polynomial for synthetic division

The next step is to adjust the polynomial we're dividing by the divisor by dividing each term by the coefficient of \(x\) in the divisor (i.e., 2). This means if our polynomial is \(p(x)\), we'll create a new polynomial \(p'(x)\) where each term of \(p(x)\) is divided by 2. This step adjusts the polynomial such that our divisor is now in the form of \(x - 2\) which can be easily used in synthetic division.

Step 3: Perform Synthetic Division

Perform synthetic division with the adjusted polynomial \(p'(x)\) and the new divisor \(x - 2\) as per the normal synthetic division process, by setting up the division and carrying out the algorithm, adding vertically and multiplying diagonally inside the synthetic division table.

Step 4: Interpret Result

From the synthetic division process, we get a quotient and a remainder, but remember, we divided the original polynomial by 2. Hence, the quotient and remainder we get is from dividing the polynomial by 2 also. Therefore, the original quotient and remainder when dividing the polynomial by \(2x - 4\) would need to be doubled to obtain the accurate result.

Key Concepts

Polynomial Division

Polynomial division is similar to long division, except it involves dividing polynomials instead of simple numbers. The goal is to determine how many times one polynomial, the divisor, is contained within another polynomial, the dividend. In this context, we typically express it as dividing a polynomial P(x) by another polynomial D(x). By doing so, we aim to find a quotient polynomial Q(x) and possibly a remainder R(x) to satisfy the equation:

• \( P(x) = D(x) \cdot Q(x) + R(x) \).

This method is crucial for simplifying polynomials or solving equations. Unlike numerical division, polynomial division begins with finding the largest term that the divisor can "fit into." This process often involves working with coefficients and powers of the variable involved in the polynomials.
Understanding polynomial division helps in gaining a clear grasp of more complex algebraic functions and their behaviors, laying the foundation for deeper studies in higher mathematics.

Coefficient Adjustment

Before applying synthetic division, especially when the divisor isn't in the simple form of \(x - c\), a crucial step involves coefficient adjustment. Coefficient adjustment refers to the process of normalizing the coefficients of the polynomials to set the divisor in a usable form for synthetic division.
If your divisor isn't neatly in the form \(x - c\), you may have to factor or divide to change it into such a form. For instance, if you're dividing by \(2x - 4\), adjust by factoring out the leading coefficient to express it as \(2(x - 2)\). This allows you to reframe the division into an effective format.

• By dividing the entire polynomial function you're dividing by, by the factor you've pulled out (like 2), you normalize the polynomial for simpler computation.
• It ensures that synthetic division aligns with the potential solutions \(c\).

This step is important to make sure that the arithmetic remains consistent and the resulting expression from the division accurately represents the original polynomial relationship.

Division Algorithm

The division algorithm is a significant mathematical concept guiding the division process. In polynomial terms, it states:

• For any polynomial \(P(x)\) and a nonzero polynomial \(D(x)\), there exist unique polynomials \(Q(x)\) (quotient) and \(R(x)\) (remainder) fulfilling \(P(x) = D(x)\cdot Q(x) + R(x)\), with \(R(x)\) having a lower degree than \(D(x)\).

This concept ensures the uniqueness and consistency of division outcomes. For synthetic division, these principles streamline calculations when dealing with polynomials with specific leading coefficients through fast computations involving only their coefficients.
Grasping the division algorithm means appreciating how every term of the dividend is systematically divided, helping to comprehend the creation of the quotient and the eventual remainder clearly.

Synthetic Division Steps

Synthetic division is a streamlined way to divide polynomials, particularly useful for divisors of the form \(x - c\). Its efficiency arises from focusing on the coefficients and using less written work than traditional long division. Follow these steps for synthetic division:

• Identify the divisor's root, \(c\).
• Adjust the polynomial if necessary by normalizing coefficients so the divisor fits the \(x - c\) pattern.
• Write down just the coefficients of the polynomial in sequence.
• Bring down the leading coefficient as it is, then multiply it by \(c\), and add it to the next coefficient. Continue this for the rest of the coefficients.
• The final number given out is the remainder.
• All previous numbers provide coefficients of the quotient polynomial.

Understanding each step of synthetic division makes it easier to solve polynomial division problems swiftly and with fewer mistakes, streamlining what can often be a tedious process into a simple, methodical calculation.