Problem 36
Question
Simplify the expression if possible. $$ \frac{121-x^{2}}{x^{2}+15 x+44} $$
Step-by-Step Solution
Verified Answer
The simplified form of the given expression is \(\frac{11-x}{x+4}\).
1Step 1: Factorize the numerator
Express \(121-x^2\) as a difference of two squares. This can be written as \((11-x)(11+x)\).
2Step 2: Factorize the denominator
Factorize \(x^2+15x+44\) which can be written as \((x+4)(x+11)\).
3Step 3: Simplify the expression
Now, plug in these factorized forms into the original expression. This results in the expression \(\frac{(11-x)(11+x)}{(x+4)(x+11)}\). Recognize and cancel the common factors, in this case \((11+x)\), from the numerator and denominator. After the cancellation, the simplified expression is \(\frac{11-x}{x+4}\).
Key Concepts
Factoring PolynomialsDifference of Two SquaresSimplifying Rational Expressions
Factoring Polynomials
Understanding how to manipulate and simplify algebraic expressions often begins with mastering the skill of factoring polynomials. Factoring is the process of breaking down a complex expression into simpler parts or 'factors' that, when multiplied together, give you the original polynomial. This is somewhat analogous to breaking down a multiplication problem into its times table components—to make it more manageable.
Consider this analogy; if you have a product like 6, which can be expressed as the product of 2 and 3, factoring polynomials operates under a similar principle but with algebraic expressions. The most common methods include finding a greatest common factor (GCF), factoring by grouping, using the difference of squares, and factoring trinomials. For instance, a polynomial like \(x^2 + 5x + 6\) would factor into \(x + 2\) and \(x + 3\), since \(x + 2\) and \(x + 3\) are the more straightforward expressions that yield the original polynomial when multiplied.
Always start by looking for a GCF before moving to other methods, which can save time and simplify the problem significantly. In the world of algebra, factoring is akin to finding a puzzle's missing pieces, and once those pieces are in place, the larger picture becomes a whole lot clearer.
Consider this analogy; if you have a product like 6, which can be expressed as the product of 2 and 3, factoring polynomials operates under a similar principle but with algebraic expressions. The most common methods include finding a greatest common factor (GCF), factoring by grouping, using the difference of squares, and factoring trinomials. For instance, a polynomial like \(x^2 + 5x + 6\) would factor into \(x + 2\) and \(x + 3\), since \(x + 2\) and \(x + 3\) are the more straightforward expressions that yield the original polynomial when multiplied.
Always start by looking for a GCF before moving to other methods, which can save time and simplify the problem significantly. In the world of algebra, factoring is akin to finding a puzzle's missing pieces, and once those pieces are in place, the larger picture becomes a whole lot clearer.
Difference of Two Squares
A special and frequently encountered factoring method is recognizing the 'difference of two squares.' The difference of two squares formula states that for any two terms \(a\) and \(b\), if you have \(a^2 - b^2\), it can be factored into \(a + b\) and \(a - b\). This works because when you multiply \(a + b\) by \(a - b\), the middle terms cancel out, and you're left with \(a^2 - b^2\) again.
For example, consider the expression \(x^2 - 9\). This expression is a difference of two squares because \(x^2\) is the square of \(x\) and 9 is the square of 3. Therefore, it factors into \(x + 3\) and \(x - 3\). Recognizing this pattern can make factoring certain polynomials almost instantaneous. But remember, this only applies when you are subtracting one square from another—hence the 'difference' in the term. Being alert to this pattern makes short work of otherwise intimidating algebra problems.
For example, consider the expression \(x^2 - 9\). This expression is a difference of two squares because \(x^2\) is the square of \(x\) and 9 is the square of 3. Therefore, it factors into \(x + 3\) and \(x - 3\). Recognizing this pattern can make factoring certain polynomials almost instantaneous. But remember, this only applies when you are subtracting one square from another—hence the 'difference' in the term. Being alert to this pattern makes short work of otherwise intimidating algebra problems.
Simplifying Rational Expressions
Once you have factored polynomials, you can apply these skills to simplify rational expressions. A rational expression is a ratio of two polynomials, similar to a fraction. Simplifying these expressions often involves factoring both the numerator and the denominator and then reducing them by canceling out common factors.
The key is to factor completely before attempting to simplify. Only factors can be canceled, not terms, so remember that term cancellation is not valid in polynomial division. For instance, after you factor the numerator and the denominator separately, you might find that they share common factors. If they do, you can simplify the expression by eliminating these shared factors. It's like reducing a fraction: if you have \( \frac{4}{8} \) you can divide both numerator and denominator by 4 to simplify it to \( \frac{1}{2} \) . The same approach applies to more complex algebraic fractions because, at their core, rational expressions follow the same rules as numeric fractions.
Simplified rational expressions are easier to work with, especially when you're dealing with equations or further algebraic manipulations. Just as simplifying a fraction can clarify a proportion, simplifying a rational expression can often clarify otherwise complicated algebraic relationships.
The key is to factor completely before attempting to simplify. Only factors can be canceled, not terms, so remember that term cancellation is not valid in polynomial division. For instance, after you factor the numerator and the denominator separately, you might find that they share common factors. If they do, you can simplify the expression by eliminating these shared factors. It's like reducing a fraction: if you have \( \frac{4}{8} \) you can divide both numerator and denominator by 4 to simplify it to \( \frac{1}{2} \) . The same approach applies to more complex algebraic fractions because, at their core, rational expressions follow the same rules as numeric fractions.
Simplified rational expressions are easier to work with, especially when you're dealing with equations or further algebraic manipulations. Just as simplifying a fraction can clarify a proportion, simplifying a rational expression can often clarify otherwise complicated algebraic relationships.
Other exercises in this chapter
Problem 36
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