Problem 10 - Beginning and Intermediate Algebra

Question

Divide. $$\left(-32 q^{6}-8 q^{3}+4 q^{2}\right) \div\left(-4 q^{2}\right)$$

Step-by-Step Solution

Answer

The simplified expression for \(\left(-32 q^{6}-8 q^{3}+4 q^{2}\right) \div\left(-4 q^{2}\right)\) is \(8q^4 + 2q -1\).

Step 1: Identify the terms of the polynomial and the monomial

We need to divide the polynomial \(-32q^6 - 8q^3 + 4q^2\) by the monomial \(-4q^2\).

Step 2: Divide each term of the polynomial by the monomial

Now, we will divide each term of the polynomial by the monomial: 1. \(\frac{-32q^6}{-4q^2}\) 2. \(\frac{-8q^3}{-4q^2}\) 3. \(\frac{4q^2}{-4q^2}\)

Step 3: Perform the divisions

We will now perform the divisions: 1. \(\frac{-32q^6}{-4q^2} = \frac{-32}{-4} \cdot \frac{q^6}{q^2} = 8q^{(6-2)} = 8q^4\) 2. \(\frac{-8q^3}{-4q^2} = \frac{-8}{-4} \cdot \frac{q^3}{q^2} = 2q^{(3-2)} = 2q^1 = 2q\) 3. \(\frac{4q^2}{-4q^2} = \frac{4}{-4} \cdot \frac{q^2}{q^2} = -1\)

Step 4: Combine the terms to obtain the simplified expression

Now we will combine the results of the divisions into one expression: \[8q^4 + 2q -1\] So, the expression \(\left(-32 q^{6}-8 q^{3}+4 q^{2}\right) \div\left(-4 q^{2}\right)\) simplifies to \(8q^4 + 2q -1\).

Key Concepts

Monomial Division

Monomial division is a process of dividing each term of a polynomial by a single monomial. It might sound complex, but it's actually quite simple if you take it step by step.
When dividing a polynomial like \(-32q^6 - 8q^3 + 4q^2\) by a monomial like \(-4q^2\), you break it down into separate divisions.
You go through each term of the polynomial:

• Divide the coefficients: Change the signs if needed and divide the numbers.
• Subtract the exponents of like bases, such as with \(q^6 \) and \(q^2 \).

Following these steps, you'll find the division becomes simple. This process simplifies complicated expressions into more manageable forms.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form by eliminating unnecessary terms and combining like terms.
Once you've divided each term of the polynomial by the monomial, you combine the results to create a simplified expression.
This involves rewriting terms like \(8q^4 + 2q - 1\) from individual parts, each a result of separate divisions.
The steps ensure that each operation follows algebraic rules, helping to get rid of complicated expressions and making them easier to understand and use in further calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra. In our example, \(-32q^6 - 8q^3 + 4q^2\) represents an algebraic expression.
When working with such expressions, it's important to understand:

• Variables, like \(q\), which can stand in for unknown or changeable numbers.
• Coefficients, the numbers multiplying the variables.
• The operations, such as addition, subtraction, multiplication, and division.

Understanding these components lets you manipulate algebraic expressions, whether by expanding, factoring, or simplifying, as demonstrated in the division process. They are central to solving equations and inequalities in algebra.