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Chapter 2: Problem 39

Divide the polynomial by the linear factor with synthetic division. Indicate the quotient \(Q(x)\) and the remainder \(r(x)\). $$\left(2 x^{4}-3 x^{3}+7 x^{2}-4\right) \div\left(x-\frac{2}{3}\right)$$

Short Answer

Expert verified
Quotient: \(Q(x) = 2x^3 - \frac{5}{3}x^2 + \frac{53}{9}x + \frac{106}{27}\); Remainder: \(r(x) = -\frac{436}{81}\).

Step by step solution

01

Setup for Synthetic Division

Identify the root of the divisor for synthetic division. Since our divisor is \(x - \frac{2}{3}\), our root is \(\frac{2}{3}\). Write down the coefficients of the polynomial \(2x^4 - 3x^3 + 7x^2 + 0x - 4\), filling in any missing terms with a zero. Thus, we have the coefficients: 2, -3, 7, 0, -4.
02

Begin Synthetic Division Process

Write the root \(\frac{2}{3}\) to the left and the coefficients (2, -3, 7, 0, -4) to the right in a row. Bring down the leading coefficient (2) to start the process.
03

Perform Division Iterations

First, multiply the root \(\frac{2}{3}\) by the value just brought down (2), which is \(\frac{4}{3}\). Add \(\frac{4}{3}\) to the next coefficient (-3), resulting in \(-\frac{5}{3}\). Repeat: multiply \(\frac{2}{3}\) by \(-\frac{5}{3}\) yielding \(-\frac{10}{9}\), add to 7 giving \(\frac{53}{9}\). Continue this process until you've processed all coefficients.
04

Final Calculation for Quotient and Remainder

Finish cascading through the coefficients: Multiply \(\frac{2}{3}\) by \(\frac{53}{9}\), resulting in \(\frac{106}{27}\), adding 0 gives \(\frac{106}{27}\). Finally, multiply \(\frac{2}{3}\) by \(\frac{106}{27}\), get \(\frac{212}{81}\), add to -4 resulting in \(-\frac{436}{81}\). Interpret the last number as the remainder.
05

Determine Quotient Polynomial

The final row of numbers forms the coefficients of the quotient polynomial. They are: \(2, -\frac{5}{3}, \frac{53}{9}, \frac{106}{27}\). So, the quotient polynomial \(Q(x) = 2x^3 - \frac{5}{3}x^2 + \frac{53}{9}x + \frac{106}{27}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is a technique used to divide one polynomial by another. It's similar to the way we perform division with numbers. However, with polynomials, we look at dividing terms based on their degrees and coefficients. There are several methods of polynomial division: synthetic division and long division are the most popular ones for specific cases.
In synthetic division, which we use when dividing a polynomial by a linear factor of the form \(x - c\), it's faster and more straightforward compared to long division. This approach makes use of the root of the divisor \(x - c\), meaning you take the opposite sign of \(c\), to divide the polynomial.
This method is particularly useful in solving large polynomial problems more quickly and efficiently, providing detailed insights for finding the quotient and remainder, crucial in further algebraic operations.
Quotient and Remainder
In polynomial division, similar to arithmetic division, each division will result in a quotient and a remainder. These two components are key outcomes of the division process and tell us about how the polynomial components relate.
The quotient, \(Q(x)\), is the polynomial we get when we divide the original polynomial without considering the remainder. It represents what fits entirely into the dividend when divided by the divisor. On performing synthetic division, these quotient coefficients are worked out systematically.
The remainder, \(r(x)\), is what is left over after dividing. In our exercise involving synthetic division, once we perform the entire process, the last number down the row represents our remainder. Here, it's crucial to note that if the remainder is zero, \(x - c\) is a factor of the polynomial, indicating that \(c\) is a root of the polynomial.
Roots of Polynomials
The roots of a polynomial are the values of \(x\) for which the polynomial equals zero. In simpler terms, they are the "solutions" to the polynomial equation \(P(x) = 0\).
Finding the roots is essential in understanding the polynomial's behavior, such as when it crosses the \(x\)-axis on a graph. When using synthetic division, if the remainder is zero, then the divisor \(x - c\) is a factor of the polynomial, and \(c\) is a root. This aspect helps in breaking down polynomials into manageable parts, especially for higher-degree polynomials.
Knowing the roots can also significantly aid in graphing, solving, or is even crucial in other mathematical concepts, such as in calculus where knowing where a function crosses the axis aids in integration and differentiation.

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