91Ó°ÊÓ

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into simpler factors that, when multiplied together, give the original polynomial. Factoring can make simplification and solving easier. In our exercise, we first need to factor the numerators and denominators of the given fractions.
For example, the numerator of the first fraction, \(28x^6 + 42x^5\), can be factored by taking out the greatest common factor (GCF). Here the GCF is \(14x^5\), so:\
\[ 28x^6 + 42x^5 = 14x^5(2x + 3) \]
For the second fraction's numerator, \(x^2 - 1\), recognize it as a difference of squares and factor it as:\
\[ x^2 - 1 = (x + 1)(x - 1) \]
Now for the denominators: The first denominator, \(x^3 - x^2\), factors as follows by taking out the common factor of \(x^2\):\
\[ x^3 - x^2 = x^2(x - 1) \]
The second denominator, \(42x + 63\), has the GCF of \(21\), so we factor it as:\
\[ 42x + 63 = 21(2x + 3) \]

After factoring, the next step in simplification is canceling common factors in the numerators and denominators. This process is similar to reducing fractions in basic arithmetic. To cancel common factors:

• Identify terms or factors in both the numerator and denominator that are the same.
• Divide them out of both to simplify the expression.

For our exercise:\
The factored form is:\
\[ \frac{14x^5(2x + 3)}{x^2(x - 1)} \times \frac{(x + 1)(x - 1)}{21(2x + 3)} \]
You can see common factors \(2x + 3\), \(x - 1\), and the GCF 7 in 14 and 21. Cancel these out as follows:
\(2x + 3\) in the numerator and denominator.
\(x - 1\) in the numerator and denominator.
The constants: \(14 / 21 = 2/3\).
This results in:
\[ \frac{14x^5}{x^2} \times \frac{x + 1}{21} = \frac{2x^5}{x^2} \times \frac{x + 1}{3} \]

The final step is to simplify the remaining algebraic expressions. Here, we combine what is left after canceling the common factors. Our expression is now:\
\[ \frac{14x^5}{x^2} \times \frac{x + 1}{21} \]
Simplify \( \frac{14x^5}{x^2}\) by subtracting the exponents since \( x^2\) divides into \( x^5\) giving \( x^{5-2}=x^3\). Thus, we have:
\[ \frac{14x^3}{21} \times (x + 1) \]
Simplify \( \frac{14}{21}\) to \( \frac{2}{3}\). So the expression simplifies to:
\[ \frac{2x^3}{3} \times (x + 1) \]
Combining terms, the final simplified expression is:
\[ \frac{2x^3(x + 1)}{3} \]
By following these steps, you can handle complex algebraic fractions by breaking them down methodically. Always factor first, cancel out common elements, and then simplify the remaining expression. These skills will be highly useful for various algebra problems you encounter.