Problem 64 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{4 a^{3} b-4 a^{2} b^{2}}{15 a-10} \cdot \frac{3 a-2}{4 a b-2 b^{2}} $$

Step-by-Step Solution

Answer

Given fractions: $$ \frac{4a^3b - 4a^2b^2}{15a - 10} \text{ and } \frac{3a - 2}{4ab - 2b^2} $$ Answer: The simplified product of the given algebraic fractions is: $$ \frac{2a( a^2 - b^2)}{5(2a - b)} $$

Step 1: Simplify fractions

First, we'll factor out any common factors in each fraction's numerator and denominator, and then simplify each fraction by cancelling out the common factors. For the first fraction: $$ \frac{4a^3b - 4a^2b^2}{15a - 10} = \frac{4ab(a^2 - b^2)}{5(3a - 2)} $$ For the second fraction: $$ \frac{3a - 2}{4ab - 2b^2} = \frac{1}{b}\frac{3a - 2}{2(2a - b)} $$

Step 2: Perform multiplication

Now that both fractions are simplified, we can perform the multiplication as follows: $$ \frac{4ab(a^2 - b^2)}{5(3a - 2)} \cdot \frac{1}{b}\frac{3a - 2}{2(2a - b)} $$

Step 3: Simplify the product

To simplify the product of the two fractions, we will first cancel out common factors in the numerator and denominator. We can see that \((3a - 2)\) and \(b\) are common factors in the numerator and denominator: $$ \frac{4a\cancelto{a^2 - b^2}{(a^2 - b^2)}}{5\cancelto{3a - 2}{(3a - 2)}} \cdot \frac{1}{\cancelto{b}{b}}\frac{\cancelto{3a - 2}{3a - 2}}{2(2a - b)} $$ Then, we multiply the fractions to get the final result: $$ \frac{4a(a^2 - b^2)}{5} \cdot \frac{1}{2(2a - b)} = \frac{4a( a^2 - b^2)}{10(2a - b)} $$ The final solution is: $$ \frac{2a( a^2 - b^2)}{5(2a - b)} $$

Key Concepts

Simplifying Fractions

When simplifying fractions, the goal is to reduce them to their simplest form. This means making sure there are no common factors remaining in the numerator and the denominator.
You can do this by factoring out any common elements first.

• Look for common factors in both the numerator and the denominator.
• Once identified, factor them out and simplify.

For example, in the original problem, the first fraction had a numerator of \(4a^3b - 4a^2b^2\) and a denominator of \(15a - 10\).
We factored it as \( \frac{4ab(a^2 - b^2)}{5(3a - 2)} \). This required pulling out the greatest common factor from both parts.
Similarly, for the second fraction, we had \( \frac{3a - 2}{4ab - 2b^2} \), which simplifies to \( \frac{1}{b} \frac{3a - 2}{2(2a - b)} \).
This involves recognizing that \(b\) and numbers in parentheses could be factored out.
By simplifying now, it makes subsequent operations easier.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler terms, or factors, that when multiplied give back the original polynomial.
This technique is indispensable when simplifying expressions, adding, subtracting, or solving equations.Factors are usually variables, numbers, or expressions that multiply to create the polynomial.
For instance, in the polynomial \(4a^3b - 4a^2b^2\), you can first factor out the common term \(4ab\), giving us \(4ab(a^2 - b^2)\).
Next, you recognize \(a^2 - b^2\) as a difference of squares, which factors further to \((a-b)(a+b)\).In the second fraction, the denominator \(4ab - 2b^2\) can be factored into \(2b(2a - b)\).
By learning and utilizing these factoring techniques, you can greatly simplify the complexity of algebraic expressions and facilitate easier calculation in tasks such as multiplication and division of polynomials.

Algebraic Multiplication

In algebraic multiplication, you're multiplying terms or whole expressions together. This involves directly multiplying numerators together and denominators together in the case of fractions.

• Multiply the coefficients (numerical factors).
• Apply the rules of exponents when multiplying variables.

In our exercise, after simplifying each fraction, we multiply:\[ \frac{4ab(a^2 - b^2)}{5(3a - 2)} \times \frac{1}{b}\frac{3a - 2}{2(2a - b)} \]First, multiply across the numerators:

• Multiply \(4ab(a^2 - b^2)\) by \(1\cdot (3a-2)\), simplifying alongside.

And across the denominators:

• Multiply \(5(3a - 2)\) by \(2b(2a - b)\).

Notice the strategic cancellation of common terms \((3a - 2)\) to simplify the process.

Algebraic Division

Algebraic division is conceptually understanding when manipulating fractions. It involves dividing algebraic expressions by a number, variable, or another expression.
When it comes to fractions, division is often simplified by multiplication with a reciprocal.In our example, we convert the expression by multiplying with the reciprocal of the second term:\[ \frac{1}{b}\frac{3a - 2}{2(2a - b)} \to \frac{2(2a - b)}{3a - 2} \]This means flipping the second fraction, making the operation a multiplication problem.
The advantage with this is it allows you to treat the operation as an extension of multiplication, preserving consistency in calculations.By viewing division in this way, we reduce the potential for error and maintain a streamlined approach to solving algebraic fraction problems.