Problem 63 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{8 x^{9}-32 x^{6}}{4 x^{4}} $$

Step-by-Step Solution

Answer

The solution is \( 2x^5 - 8x^2 \).

Step 1: Simplify the Numerator

Look at the numerator \( 8x^9 - 32x^6 \). Factoring out the greatest common factor, which is \( 8x^6 \), gives \( 8x^6(x^3 - 4) \).

Step 2: Divide Each Term

Now, divide each term in the simplified numerator by the denominator \( 4x^4 \). Start by dividing \( 8x^6(x^3 - 4) \):\[ \frac{8x^6 imes x^3}{4x^4} - \frac{8x^6 imes 4}{4x^4} \].

Step 3: Perform the Division

Now, perform the division for each term:- \( \frac{8x^9}{4x^4} = 2x^5 \)- \( \frac{32x^6}{4x^4} = 8x^2 \).

Step 4: Write the Result

The expression simplifies to:\[ 2x^5 - 8x^2 \].

Key Concepts

Factoring Polynomials

Factoring polynomials is essentially the process of breaking down a polynomial into simpler terms, or "factors," which can, when multiplied together, give the original polynomial.
In mathematics, some polynomials can be factored into multiple polynomials of lower degrees, showing more simplicity and revealing relationships hidden in the original expression.

For instance, take the expression from our exercise: \[ 8x^9 - 32x^6 \]. Here, we identified that the greatest number and variable shared across both terms is \( 8x^6 \). By factoring out \( 8x^6 \), you simplify the expression to: \[ 8x^6(x^3 - 4) \].
This technique allows you to see the overall structure and to facilitate operations such as division more efficiently.

• Key concept: Simplification by factoring makes operations like division more manageable.
• Helps in understanding polynomial relationships and potential symmetrical properties.
• Finding the greatest common factor is a crucial initial step in polynomial factoring.

Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that divides each of the terms without leaving a remainder.
This is a useful tool in simplifying polynomial expressions.
Let's explore how finding the GCF of polynomial expressions, like the example in our problem, can simplify operations.

In the operation \( 8x^9 - 32x^6 \), we identified that both terms are divisible by \( 8x^6 \):

• The number \( 8 \) is the greatest common factor of the numeric coefficients.
• The variable \( x^6 \) is the greatest common term since it is the highest power of x that divides both terms evenly.

By factoring out \( 8x^6 \), it enables easier manipulation and helps simplify further algebraic operations.
Finding the GCF is a foundational skill in algebra to simplify complex polynomial expressions and solve polynomial equations effectively.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols.
These expressions form the building blocks of algebra, allowing us to create and manipulate mathematical models that represent real-world scenarios or abstract mathematical problems.

In our example, one such expression is: \( \frac{8x^9 - 32x^6}{4x^4} \). The structure shows a division format between two algebraic expressions.

• The numerator: \( 8x^9 - 32x^6 \), which was simplified using factoring techniques.
• The denominator: \( 4x^4 \), which is necessary for the division step in our expression.

Understanding these elements is vital in manipulating and simplifying algebraic expressions, as they form the core of many types of problems in algebra.
Whether solving equations, performing function operations, or breaking down complex models, grasping how to work with algebraic expressions ensures a deeper comprehension and ease in mathematical problem-solving.