Problem 72 - Algebra 2

Question

PREREQUISITE SKILL Find each quotient. $$ \left(4 x^{3}-8 x^{2}-5 x-10\right) \div(x+2) $$

Step-by-Step Solution

Answer

Quotient is \(4x^2 - 16x + 27\) with remainder \(-64\).

Step 1: Setup Synthetic Division

Since we are dividing the polynomial \(4x^3 - 8x^2 - 5x - 10\) by \(x + 2\), we will use synthetic division. Start by writing down the coefficients of the polynomial: 4, -8, -5, -10. Because we divide by \(x + 2\), we use \(-2\) as our synthetic divisor.

Step 2: Perform Synthetic Division

Write the synthetic divisor \(-2\) to the left and the coefficients \(4, -8, -5, -10\) to the right separated by a vertical bar. Bring down the first coefficient, which is 4, as is.- Multiply \(-2\) by 4 and write the result, \(-8\), under the next coefficient.- Add \(-8\) to \(-8\) to get \(-16\).- Multiply \(-2\) by \(-16\) to get 32.- Add 32 to \(-5\) to get 27.- Multiply \(-2\) by 27 to get \(-54\).- Add \(-54\) to \(-10\) to get \(-64\). The row you obtained, 4, -16, 27, \(-64\), gives the quotient coefficients and the remainder.

Step 3: Interpret Results

The result from synthetic division gives the coefficients of the quotient polynomial and a remainder. The values \(4, -16, 27\) are the coefficients of the quotient polynomial, starting with the degree one less than the original polynomial. The \(-64\) is the remainder. Thus, \(4x^2 - 16x + 27\) is the quotient, and \(-64\) is the remainder.

Step 4: Express Final Solution

The division of \(4x^3 - 8x^2 - 5x - 10\) by \(x + 2\) results in a quotient of \(4x^2 - 16x + 27\) with a remainder of \(-64\). This can be expressed as:\[\frac{4x^3 - 8x^2 - 5x - 10}{x + 2} = 4x^2 - 16x + 27 - \frac{64}{x + 2}\]

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, often simpler, polynomial. It's similar to long division but with variables. In this process, we find a quotient and sometimes a remainder, just like in arithmetic division.

When performing polynomial division, there are two main techniques you can use:

• Long Division: This is similar to regular long division used with numbers, but involves algebraic expressions.
• Synthetic Division: A shorthand method used specifically when dividing by linear polynomials of the form \( x - c \).

Choosing between these methods depends on the polynomial you are working with. Synthetic division is quicker and often used when the divisor is a simple binomial like \(x + 2\).

By understanding these processes, you can simplify complex algebraic expressions and solve higher-degree polynomial equations efficiently.

Quotient and Remainder

In polynomial division, just like with numbers, dividing doesn't always result in a clean quotient. Sometimes there is a remainder. To understand this concept, think of dividing 10 by 3, which gives a quotient of 3 and a remainder of 1.

In our example, we use synthetic division to divide \(4x^3 - 8x^2 - 5x - 10\) by \(x + 2\). The result shows a quotient of \(4x^2 - 16x + 27\) and a remainder of \(-64\).

The general formula we use is:\[\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}\]For our exercise: \[4x^3 - 8x^2 - 5x - 10 = (x + 2)(4x^2 - 16x + 27) - 64\]

Knowing how to identify the quotient and remainder is crucial for simplifying expressions and solving polynomial equations. It helps us express the result of division more clearly.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators such as addition, subtraction, multiplication, and division. These expressions form the basis of algebra and are used to represent real-world situations mathematically.

Algebraic expressions are versatile:

• They can represent simple numbers, or more complex entities like polynomials.
• Polynomials themselves are algebraic expressions that consist of variables and coefficients.

In our exercise, expressions like \(4x^3 - 8x^2 - 5x - 10\) and \(x + 2\) are polynomials.

Manipulating and simplifying these expressions through operations like polynomial division allows us to break down complex problems into simpler parts. This is crucial in algebra, helping us uncover unknown values and understand relationships between variables.