Problem 68 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{k^{2}-36}{k+3} \div \frac{k+6}{-k-3} $$

Step-by-Step Solution

Answer

-(k - 6)

Step 1: Factor the Numerator

Factor the numerator of the first fraction. Notice that the numerator is a difference of squares: \[ k^2 - 36 = (k + 6)(k - 6) \]. Replace the numerator with its factored form.

Step 2: Simplify the Fraction

Rewrite the expression after factoring: \[ \frac{(k + 6)(k - 6)}{k + 3} \frac{k+6}{-k-3} \]

Step 3: Division of Fractions

Change the division to multiplication and take the reciprocal of the second fraction: \[ \frac{(k + 6)(k - 6)}{k + 3} \times \frac{-k-3}{k+6} \]

Step 4: Simplify the Reciprocal

Simplify the reciprocal of the second fraction: \[ \frac{-k-3}{k+6} = \frac{-(k + 3)}{k + 6} \]. Now the expression looks like this: \[ \frac{(k + 6)(k - 6)}{k + 3} \times \frac{-(k + 3)}{k + 6} \]

Step 5: Cancel Common Factors

Cancel the common factors in the numerator and the denominator: \[ \frac{(k + 6) \textcolor{red}{(k - 6)}}{\textcolor{red}{k + 3}} \times \frac{\textcolor{red}{-(k + 3)}}{\textcolor{red}{k + 6}} \]. We are left with: \[ -(k - 6) \]

Key Concepts

Factoring Polynomials

When simplifying algebraic fractions, one critical step is factoring polynomials. In the given exercise, the expression \( k^2 - 36 \) is a polynomial that can be factored to simplify the fraction. Factoring involves writing the polynomial as a product of its factors. For \( k^2 - 36 \), we recognize it as a difference of squares. The formula for a difference of squares is \[ a^2 - b^2 = (a + b)(a - b) \]. Applying this to \( k^2 - 36 \), where \( a = k \) and \( b = 6 \), we get \[ k^2 - 36 = (k + 6)(k - 6) \]. Factoring simplifies complex expressions and makes it easier to cancel common factors later.

Difference of Squares

The difference of squares is a special type of factoring in polynomials. It occurs when we subtract one square number from another. In the exercise, the expression \( k^2 - 36 \) is a classic example. The general form is \[ a^2 - b^2 \], and it factors into \[ (a + b)(a - b) \]. This factorization helps to break down and simplify what initially might seem like a complex polynomial. Recognizing and applying the difference of squares rule is a crucial algebraic skill that facilitates easier manipulation and simplification of expressions.

Division of Fractions

Dividing fractions is a key operation in simplifying algebraic fractions. Rather than directly dividing, we multiply by the reciprocal. In the given exercise, the division \[ \frac{(k + 6)(k - 6)}{k + 3} \div \frac{k + 6}{-k - 3} \] is converted to a multiplication problem \[ \frac{(k + 6)(k - 6)}{k + 3} \times \frac{-k - 3}{k + 6} \]. This transformation is vital because it allows us to handle the problem using multiplication techniques, making it easier to cancel common factors and simplify the expression.

Reciprocal

The reciprocal of a number or expression is what you multiply it by to get 1. For a fraction \[ \frac{a}{b} \], its reciprocal is \[ \frac{b}{a} \]. In this exercise, we need the reciprocal of the second fraction \[ \frac{k + 6}{-k - 3} \]. Finding its reciprocal gives us \[ \frac{-k - 3}{k + 6} \]. Simplifying the reciprocal can involve recognizing patterns, such as inverting negative signs to factor our expressions correctly, which in turn aids in cancellation.

Canceling Common Factors

Canceling common factors is a powerful technique used to simplify algebraic fractions. Once we have factored polynomials and written the division as multiplication, we can cancel out terms that appear in both the numerator and denominator. In the exercise, key common factors \((k + 6)\) and \((k + 3)\) are identified and canceled: \[ \frac{(k + 6)(k - 6)}{k + 3} \times \frac{-(k + 3)}{k + 6} = \frac{(\cancel{k + 6})(k - 6)}{\cancel{k + 3}} \times \frac{-(\cancel{k + 3})}{\cancel{k + 6}} \]. What remains is \(- (k - 6)\). Canceling these common factors reduces the fraction to its simplest form.