Problem 51
Question
Multiply. $$\frac{x^{2}-3 x-4}{x^{2}+6 x+5} \cdot \frac{x^{2}+5 x+6}{8+2 x-x^{2}}$$
Step-by-Step Solution
Verified Answer
\(\frac{(x+3)}{(x+5)}\)
1Step 1: Factorize the Polynomials
Factor all the expressions: \((x^{2}-3x-4) = (x-4)(x+1)\), \((x^{2}+6x+5) = (x+5)(x+1)\), \((x^{2}+5x+6) = (x+3)(x+2)\), and \((8+2x-x^{2}) = -(x-4)(x+2)\). Write the rational expressions with factored polynomials now.
2Step 2: Multiply fractions
After factoring, just follow the rule for multiplying fractions: Multiply the numerators together for the new numerator and denominators together for the new denominator. This gives: \(\frac{(x-4)(x+1)(x+3)(x+2)}{(x+5)(x+1)(x-4)(x+2)}\).
3Step 3: Simplify the Result
Cancel out common factors in the numerator and denominator. We have (x-4) and (x+2) in both places. So, we are left with: \(\frac{(x+3)}{(x+5)}\). That simplifies the multiplication of the given rational expressions.
Key Concepts
Rational ExpressionsMultiplying FractionsSimplifying Algebraic Expressions
Rational Expressions
Rational expressions are like fractions, but instead of numbers in the numerator and denominator, we have polynomials. A polynomial is just a fancy term for expressions with variables raised to whole number powers, like \(x^2 - 3x - 4\) or \(x^2 + 6x + 5\).
Understanding rational expressions is crucial when dealing with algebraic fractions. Just like regular fractions, they can be simplified, multiplied, or divided.
When simplifying rational expressions, you'll often need to factorize the polynomials first. This means expressing the polynomial as a product of simpler expressions, which makes it easier to identify and cancel out common factors, much like reducing fractions.
One of the most important steps in handling rational expressions is to ensure that the denominator never equals zero, as division by zero is undefined. Always check your solutions or expressions to avoid this issue.
Understanding rational expressions is crucial when dealing with algebraic fractions. Just like regular fractions, they can be simplified, multiplied, or divided.
When simplifying rational expressions, you'll often need to factorize the polynomials first. This means expressing the polynomial as a product of simpler expressions, which makes it easier to identify and cancel out common factors, much like reducing fractions.
One of the most important steps in handling rational expressions is to ensure that the denominator never equals zero, as division by zero is undefined. Always check your solutions or expressions to avoid this issue.
Multiplying Fractions
Multiplying fractions, whether they are numbers or expressions, follows a straightforward rule. You multiply the numerators together to get the new numerator and the denominators together to get the new denominator.
For example, if you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), their multiplication is straightforward as \[\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}.\]
In the context of multiplying rational expressions as in our exercise, once the polynomials are factored, the same rule applies. Multiply across the top (for numerators) and across the bottom (for denominators).
After multiplying, it's essential to always simplify. Simplifying involves factoring both the numerator and the denominator and canceling any shared factors. It results in a more concise and understandable expression.
For example, if you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), their multiplication is straightforward as \[\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}.\]
In the context of multiplying rational expressions as in our exercise, once the polynomials are factored, the same rule applies. Multiply across the top (for numerators) and across the bottom (for denominators).
After multiplying, it's essential to always simplify. Simplifying involves factoring both the numerator and the denominator and canceling any shared factors. It results in a more concise and understandable expression.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is all about making them as neat and straightforward as possible, often by canceling out like terms or common factors.
For rational expressions, simplifying involves factoring the expressions both in the numerator and the denominator. This means finding two or more simpler expressions that multiply together to make the original polynomial.
Common factors can then be "canceled out" from the numerator and the denominator. Imagine you have \((x-4)\) in both the numerator and the denominator. They cancel each other out, just like how 3 cancels itself when it appears in both parts of a fraction \(\frac{3}{3}\).
Keep an eye out for the same terms or factors in both the numerator and the denominator. But remember, you only cancel terms that are factors of the whole numerator and the whole denominator, not just individual parts. This discipline ensures accuracy in simplifying expressions.
For rational expressions, simplifying involves factoring the expressions both in the numerator and the denominator. This means finding two or more simpler expressions that multiply together to make the original polynomial.
Common factors can then be "canceled out" from the numerator and the denominator. Imagine you have \((x-4)\) in both the numerator and the denominator. They cancel each other out, just like how 3 cancels itself when it appears in both parts of a fraction \(\frac{3}{3}\).
Keep an eye out for the same terms or factors in both the numerator and the denominator. But remember, you only cancel terms that are factors of the whole numerator and the whole denominator, not just individual parts. This discipline ensures accuracy in simplifying expressions.
Other exercises in this chapter
Problem 51
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