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Chapter 2: Problem 62

Factor each expression completely. $$ x^{2}+8 x+16-y^{2} $$

Short Answer

Expert verified
Answer: (x+4+y)(x+4-y)

Step by step solution

01

Identify the perfect squares

In the given expression \(x^2 + 8x + 16 - y^2\), we can see that both \(x^2\) and \(y^2\) are perfect squares, and the middle term, \(8x\), can be used to find another perfect square. We notice that \(16 = 4^2\), which means we can try factoring \((x+4)(x+4)\).
02

Factor the perfect square trinomial

We can factor \(x^2 + 8x + 16\) into \((x+4)(x+4)\) or \((x+4)^2\). So the expression becomes: $$ (x+4)^2 - y^2 $$
03

Apply the difference of squares formula

Now we have a difference of squares: \((x+4)^2 - y^2\). The difference of squares formula is given by: $$ a^2 - b^2 = (a+b)(a-b) $$ In this case, \(a = (x+4)\) and \(b=y\). Applying the formula, we get: $$ ((x+4) + y)((x+4) - y) $$
04

Final answer

The completely factored expression is: $$ (x+4+y)(x+4-y) $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial
A perfect square trinomial is a special kind of polynomial that can be factored into the square of a binomial. Imagine expanding
  • \((a + b)^2\)
  • \((a - b)^2\)
It becomes a trinomial:
  • \(a^2 + 2ab + b^2\)
  • \(a^2 - 2ab + b^2\)
In the expression \(x^2 + 8x + 16\), notice how it matches up with the perfect square trinomial structure. Here,
\(a = x\) and
\(b = 4\). Therefore, the expression
  • \(x^2 + 8x + 16\)
factors neatly into
  • \((x + 4)^2\)
    • .
Understanding perfect square trinomials can make factoring polynomials quicker and easier every time you come across them! By looking for this specific pattern, you can save time and simplify the process.
Difference of Squares
The difference of squares is a special algebraic pattern that occurs when you subtract one square number from another. The general formula is:
  • \(a^2 - b^2 = (a + b)(a - b)\)
Let's apply this to the expression
  • \((x+4)^2 - y^2\)
To see why this works, think about how squares behave. They "expand" around zero, so subtracting them yields two factors.In this case:
  • \(a = (x + 4)\)
  • \(b = y\)
Plug these into the difference of squares formula:
  • \(((x+4) + y)((x+4) - y)\)
This type of factoring is quite useful because it breaks down complex expressions quickly. Any time you see a subtraction between two squares, it鈥檚 a clue that you can use this method.
Polynomial Expressions
Polynomial expressions are mathematical expressions involving variables and coefficients. The simplest are monomials, like
  • \(x^2\)
Polynomials can grow more complex with terms like:
  • \(ax^2 + bx + c\)
The complexity arises as we combine these terms. In the exercise, we started with a polynomial
  • \(x^{2} + 8x + 16 - y^{2}\)
Polynomials can often be factored to simplify them or to solve equations. Knowing how to identify structures like perfect square trinomials and differences of squares is invaluable. Factoring turns polynomials into products of simpler expressions, revealing roots or solutions more easily. Practice leads to familiarity with patterns, accelerating the solving process in many algebra problems.

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Most popular questions from this chapter

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