Problem 120 - Introductory Algebra for College Students

Question

Factor completely. $$x^{2}+14 x+49-16 a^{2}$$

Step-by-Step Solution

Answer

The factored form of the polynomial is \((x+7+4a)(x+7-4a).\)

Step 1: Identify the perfect square trinomial

The first three terms form a perfect square trinomial, which can be expressed as \((x+7)^2 = x^2+14x+49.\)

Step 2: Rewrite the expression

By substituting into the original expression the breakdown of the perfect square trinomial identified in step 1, the polynomial can be rewritten as \((x+7)^2-16a^2\).

Step 3: Utilize difference of squares

The expression is now a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a+b)(a-b).\) Apply this formula to get the final expression as \((x+7+4a)(x+7-4a).\)

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is a special form of quadratic polynomial that can be expressed as the square of a binomial. It follows the pattern of \(a^{2} + 2ab + b^{2}\), where \(a\) and \(b\) represent any number or expression. When factored, it simplifies into \( (a + b)^{2} \), indicating that both \(a\) and \(b\) have been squared and added together, along with the product of their double.

In the given exercise, the trinomial \(x^{2} + 14x + 49\) perfectly matches this pattern, as it can be seen that \(14x\) is twice the product of \(x\) and \(7\), and \(49\) is the square of \(7\). This makes the trinomial a perfect square, factored into \( (x + 7)^{2} \). Recognizing perfect square trinomials is essential for simplifying polynomials and for further factorization.

The Difference of Squares

The difference of squares is a common mathematical concept used in factoring polynomials. It is based on the identity \(a^{2} - b^{2} = (a - b)(a + b)\). This identity tells us that any expression where two squared terms are subtracted can be factored into the product of the sum and the difference of the square roots of those terms.

Applying this concept to the solution from the exercise, after recognizing the perfect square trinomial, we observe that the polynomial can be rewritten as \( (x + 7)^{2} - 16a^{2}\). Here, \( (x + 7)^{2}\) is our \(a^{2}\), and \(16a^{2}\) is our \(b^{2}\). These are both perfect squares, and their difference allows us to factor the expression into \( (x + 7 + 4a)(x + 7 - 4a)\), demonstrating the difference of squares in action.

Steps to Polynomial Factorization

Factorization of polynomials can often seem daunting, but breaking it down into methodical steps can simplify the process. Here are the essential steps for factorization:

• Identify Special Patterns: Look for patterns such as perfect square trinomials, difference of squares, or other recognizable forms like sum of cubes or difference of cubes.
• Rewrite the Expression: Use the identified patterns to rewrite parts of the polynomial into a form that can be factored more simply.
• Apply Factorization Formulas: Use algebraic identities like \(a^{2} - b^{2} = (a - b)(a + b)\) to factor the rewritten expression.
• Simplify: Simplify the factored terms to get the final factored form of the polynomial.

Referring to our example, first, we identified the perfect square trinomial, then we rewrote the expression to highlight the difference of squares, which we further factored using the appropriate identity. This structured approach can be applied to complex polynomials, breaking them into more manageable pieces that can be factored sequentially.