Problem 119 - Introductory Algebra for College Students

Question

Factor completely. $$x^{2}+8 x+16-25 a^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the given quadratic expression \(x^{2}+8 x+16-25 a^{2}\) is \((x+4+5a)(x+4-5a)\).

Step 1: Recognize the square of a binomial

The expression can be rewritten as: \(x^{2} + 8x + 16 - 25a^{2}\) = \((x+4)^{2} - (5a)^2\). This approach is possible because the first part of the equation \(x^{2} + 8x + 16\) is a square of a binomial.

Step 2: Apply the difference of squares formula

According to the difference of squares formula, it states that for any real numbers A and B, \(A^{2} - B^{2} = (A+B)(A-B)\). We have already identified A as \(x+4\) and B as \(5a\). By applying the formula, we get: \((x+4)^{2} - (5a)^{2} = (x+4+5a)(x+4-5a)\).

Step 3: Write the factored form

After using the difference of squares formula, the original expression has been turned into a completely factored form. Therefore, \(x^{2}+8 x+16-25 a^{2} = (x+4+5a)(x+4-5a)\).

Key Concepts

Square of a Binomial

Understanding the square of a binomial is a fundamental aspect of algebra. A binomial is an algebraic expression containing two terms, such as (a + b). The square of a binomial, which is written as \((a+b)^2\), represents the binomial multiplied by itself. The resulting expanded form is \(a^2 + 2ab + b^2\). In our exercise, the term \(x^2 + 8x + 16\), upon closer inspection, reveals itself to be a perfect square of the binomial \((x+4)\).

To identify a square of a binomial, look for three terms where:

• The first and last terms are perfect squares.
• The middle term is twice the product of the roots of the first and last terms.

Recognizing this pattern allows you to quickly rewrite the trinomial as a squared binomial, simplifying the factoring process significantly.

Difference of Squares Formula

The difference of squares formula is a useful tool in algebra that describes a specific relationship between two squared terms. It states that the difference between two squares can be factored into the product of the sum and the difference of the roots. Mathematically, the formula is given by \(A^2 - B^2 = (A + B)(A - B)\).

In the context of our exercise, after recognizing that we have a square of a binomial and a separate squared term, we can see that \((x+4)^2 - (5a)^2\) represents a difference of squares. We can apply the difference of squares formula here to factor the expression into \((x+4+5a)(x+4-5a)\). This process requires that we correctly identify 'A' and 'B' from the original expression in order to factor it correctly.

Algebraic Expression

At its core, an algebraic expression is a combination of numbers, variables (like x and y), and arithmetic operations (such as add, subtract, multiply, and divide). In our exercise, \(x^2 + 8x + 16 - 25a^2\) is a complex algebraic expression that combines polynomials and a variable to the second power.

Breaking down algebraic expressions into their constituent parts can often reveal underlying patterns or structures—such as the square of a binomial or the difference of squares—as in the provided exercise. By doing this, we can not only simplify the expression but also understand the relationship between its components, which is crucial for solving algebraic equations and simplifying complex problems.

Factored Form

To understand the factored form, imagine breaking down a number into its building blocks—the prime numbers that multiply together to give the original number. Similarly, in algebra, expressing a polynomial in factored form involves decomposing the expression into a product of its factors which are often simpler polynomials or binomials. In our example, the original expression \(x^2 + 8x + 16 - 25a^2\) is transformed into \((x+4+5a)(x+4-5a)\), demonstrating its factored form and revealing its roots.

Factored form is particularly useful when solving polynomial equations, as it makes the roots—the values of the variable that make the equation true—immediately apparent. In this case, setting each factor equal to zero allows for the solving of the equation, which is a cornerstone of algebra.