Problem 70 - Elementary Algebra

Question

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{p^{3}-8}{p^{2}-4 p+4} $$

Step-by-Step Solution

Answer

\[\frac{p^2 + 2p + 4}{p - 2}\]

Step 1: Identify the type of expression

Recognize that the expression to be simplified is a rational expression with a polynomial in the numerator and another polynomial in the denominator.

Step 2: Factor the numerator

Factor the numerator, which is a difference of cubes. Recall the difference of cubes formula: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]Here, \(p^3 - 8\) can be written as \(p^3 - 2^3\). Applying the formula: \[p^3 - 8 = (p - 2)(p^2 + 2p + 4)\]

Step 3: Factor the denominator

Factor the denominator by recognizing it as a perfect square trinomial. The expression \(p^2 - 4p + 4\) matches the pattern \((a - b)^2 = a^2 - 2ab + b^2\) where \(a = p\) and \(b = 2\). So, \[p^2 - 4p + 4 = (p - 2)^2\]

Step 4: Simplify the expression

Bring together the factored forms of the numerator and the denominator: \[\frac{(p - 2)(p^2 + 2p + 4)}{(p - 2)^2}\] Notice that \(p - 2\) is a common factor in both the numerator and the denominator. Cancel out \(p - 2\) to simplify the expression: \[\frac{p^2 + 2p + 4}{p - 2}\]

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components, called factors, that when multiplied together give back the original polynomial. This is a crucial skill in simplifying algebraic expressions and solving equations.

Here are some tips:

• Look for common factors in each term of the polynomial first.
• Identify any special factoring patterns like the difference of squares, perfect square trinomials, or the sum/difference of cubes.
• Once any common factors are taken out, check for other factoring patterns or techniques like grouping.

Always make sure that the polynomial is completely factored before moving on to simplify any rational expressions.

For example, in the original exercise, the numerator was factored by recognizing it as a difference of cubes, while the denominator was factored by identifying it as a perfect square trinomial.

Difference of Cubes

The formula for the difference of cubes is very useful when simplifying polynomials and rational expressions. It states that: ewlineewlinea^3 - b^3 = (a - b)(a^2 + ab + b^2)ewlineewlinea^3 - b^3 is a special polynomial where the result is very structured. In other words, for any two numbers a and b, the expression a^3 - b^3 can always be factored using this formula.

Let’s take an example: Consider the polynomial p^3 - 8 from our exercise. By recognizing that 8 can be written as 2^3, we can apply the difference of cubes formula:

p^3 - 2^3 = (p - 2)(p^2 + 2p + 4)Now the polynomial is factored into two simpler expressions. Always remember this pattern as it simplifies many seemingly complicated problems in algebra.

Perfect Square Trinomial

A perfect square trinomial is another special factoring formula you'll frequently encounter in algebra. It has a specific form and can be factored into a square of a binomial. The general pattern for a perfect square trinomial is:ewlineewlinea^2 - 2ab + b^2 = (a - b)^2In our exercise, the denominator p^2 - 4p + 4 corresponds to this pattern. By identifying this, we can factor it into:ewlineewlinep^2 - 4p + 4 = (p - 2)^2This recognition helps simplify complex algebraic expressions by reducing them to more manageable forms. Remember, spotting these patterns means you can solve these types of problems more quickly and accurately.

By combining this factorization with the numerator’s factorization, you can then cancel out the common factors, simplifying the rational expression significantly.