Problem 52 - Elementary and Intermediate Algebra

Question

Factor by first grouping the appropriate terms. \(x^{2}+10 x+25-16 z^{2}\)

Step-by-Step Solution

Answer

The expression factors to \((x + 5 - 4z)(x + 5 + 4z)\).

Step 1: Recognize the Square of a Binomial

Look at the first three terms: \(x^2 + 10x + 25\). Notice that they form a perfect square trinomial. A perfect square trinomial can be written as \((a + b)^2\) where \(a = x\) and \(b = 5\). Therefore, \(x^2 + 10x + 25 = (x+5)^2\).

Step 2: Rewrite the Expression

Use the result from Step 1 to rewrite the entire expression: \((x + 5)^2 - 16z^2\). This expression is now in the form of a difference of squares, \(A^2 - B^2\).

Step 3: Factor the Difference of Squares

The expression \((x + 5)^2 - (4z)^2\) is a difference of squares. Recall the formula: \(A^2 - B^2 = (A - B)(A + B)\). Here, \(A = x + 5\) and \(B = 4z\), so the factorization is \((x + 5 - 4z)(x + 5 + 4z)\).

Key Concepts

Factoring

Factoring in algebra is a process of breaking down an expression into simpler, "factor" components that when multiplied together give the original expression. It is similar to splitting numbers into their prime factors but applied to algebraic terms. Factoring is a powerful tool that simplifies expressions and even solves equations.
In algebra, common techniques include taking out common factors, grouping terms, or recognizing special patterns like perfect square trinomials and differences of squares. These patterns allow for straightforward factoring and often involve using specific formulas. Understanding when and how to factor is crucial.

• Factoring makes complex algebraic expressions easier to understand and solve.
• This method is essential for solving quadratic equations amongst others.
• Recognizing special algebraic patterns expedites the factoring process.

When approaching a factoring problem, the key first step is to identify which method would be most efficient for the given expression. Sometimes a combination of techniques is the way forward, as shown in the step-by-step solution above.

Perfect Square Trinomial

A perfect square trinomial is an important algebraic pattern that is the square of a binomial. It takes the form:

• \(a^2 + 2ab + b^2 = (a + b)^2\)
• \(a^2 - 2ab + b^2 = (a - b)^2\)

Identifying a perfect square trinomial can significantly simplify factoring. As seen in the problem, the expression \(x^2 + 10x + 25\) is recognized as a perfect square trinomial since:

• The first term \(x^2\) and the last term \(25\) are perfect squares.
• The middle term \(10x\) is twice the product of \(x\) and \(5\).

Thus, the trinomial \(x^2 + 10x + 25\) can be rewritten as \((x + 5)^2\). Understanding and spotting this form early on simplifies factoring into simpler expressions, making it an essential skill in handling algebraic expressions.

Difference of Squares

Difference of squares is another crucial pattern in algebra and is characterized by the expression \(A^2 - B^2\). This pattern is factorable using the formula:

• \(A^2 - B^2 = (A - B)(A + B)\)

In the given problem, the rewritten expression \((x + 5)^2 - 16z^2\) is in the format of a difference of squares because:

• \((x + 5)^2\) is a perfect square.
• \(16z^2\), or \((4z)^2\), is also a perfect square.

The expression can be factored as \((x + 5 - 4z)(x + 5 + 4z)\). Understanding and recognizing this pattern is very useful in simplifying expressions and solving equations efficiently. Remember that whenever two squares are subtracted, the difference of squares method can always be applied, which simplifies complex expressions quickly.