Textbook Question
In Exercises 15–32, multiply or divide as indicated. (x2+x)/(x2−4) ÷ (x2−1)/(x2+5x+6)
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0. Review of Algebra
Polynomials Intro
2:56 minutesProblem 61
Textbook Question
In Exercises 33–68, add or subtract as indicated. 4/(x2+6x+9) + 4/(x+3)
Verified step by step guidance1
Step 1: Recognize that the denominators in the given expression are different. The first term has a denominator of \(x^2 + 6x + 9\), and the second term has a denominator of \(x + 3\). Factorize \(x^2 + 6x + 9\) to simplify the expression. Notice that \(x^2 + 6x + 9\) is a perfect square trinomial, so it can be rewritten as \((x + 3)^2\).
Step 2: Rewrite the expression using the factored form of the first denominator: \(\frac{4}{(x + 3)^2} + \frac{4}{x + 3}\). Now, observe that the denominators are \((x + 3)^2\) and \(x + 3\), which are not the same. To combine these fractions, we need a common denominator.
Step 3: Determine the least common denominator (LCD). The LCD for \((x + 3)^2\) and \(x + 3\) is \((x + 3)^2\). Rewrite the second fraction \(\frac{4}{x + 3}\) so that it has the LCD as its denominator. Multiply both the numerator and denominator of \(\frac{4}{x + 3}\) by \(x + 3\), resulting in \(\frac{4(x + 3)}{(x + 3)^2}\).
Step 4: Combine the two fractions under the common denominator \((x + 3)^2\). The first fraction remains \(\frac{4}{(x + 3)^2}\), and the second fraction becomes \(\frac{4(x + 3)}{(x + 3)^2}\). Add the numerators: \(4 + 4(x + 3)\), keeping the denominator \((x + 3)^2\).
Step 5: Simplify the numerator. Distribute the \(4\) in \(4(x + 3)\) to get \(4x + 12\). Combine this with the \(4\) from the first term to get \(4x + 16\). The final expression is \(\frac{4x + 16}{(x + 3)^2}\).
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Quadratics
Factoring quadratics involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. In the given expression, x^2 + 6x + 9 can be factored into (x + 3)(x + 3) or (x + 3)^2. This simplification is crucial for combining fractions with a common denominator.
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Finding a Common Denominator
To add or subtract fractions, it is essential to have a common denominator. In this case, the denominators are (x^2 + 6x + 9) and (x + 3). Since (x + 3) is a factor of (x^2 + 6x + 9), the common denominator can be identified as (x + 3)^2, allowing for the fractions to be combined effectively.
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Simplifying Rational Expressions
Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. After finding a common denominator and combining the fractions, it is important to simplify the resulting expression to make it easier to interpret and work with.
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