Problem 107 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(x^{2}+16 x+64\)

Step-by-Step Solution

Answer

The expression factors as \((x + 8)^2\).

Step 1: Identify the Expression Type

The given expression is a quadratic trinomial of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 16 \), and \( c = 64 \).

Step 2: Check for Perfect Square Trinomial

To determine if the quadratic is a perfect square trinomial, check if \( b^2 = 4ac \). Substitute the values to get \( 16^2 = 4 \times 1 \times 64 \). Simplifying, \( 256 = 256 \), confirming it is a perfect square trinomial.

Step 3: Find the Square Roots

Since the quadratic is a perfect square trinomial, we can find the number that, when squared, equals \( a \) and \( c \). The square root of \( a = 1 \) is 1, and the square root of \( c = 64 \) is 8.

Step 4: Factor the Trinomial

Using the fact that the expression is a perfect square trinomial, rewrite the expression as a binomial squared: \((x + 8)^2\).

Step 5: Verify the Factorization

Expand \((x + 8)^2\) to ensure it equals the original expression: \((x + 8)(x + 8) = x^2 + 8x + 8x + 64 = x^2 + 16x + 64\). The factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic polynomial. It results when a binomial is squared, meaning it is the product of multiplying a binomial by itself. In simpler terms, if you square a binomial, you will get a perfect square trinomial. For example, if you have \((x + a)^2\), expanding it results in \(x^2 + 2ax + a^2\). This expanded form is a perfect square trinomial.
A quick way to check if a quadratic trinomial is a perfect square is to calculate \(b^2\) and see if it equals \(4ac\), where \(ax^2 + bx + c\) is the original trinomial. In the given expression \(x^2 + 16x + 64\), substituting the values gives \(16^2 = 256\) and \(4 \times 1 \times 64 = 256\), showing that the expression is indeed a perfect square trinomial.

Quadratic Trinomial

A quadratic trinomial is an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. This class of polynomials is fundamental in algebra because they form the basis of quadratic equations and functions. Recognizing the standard form is crucial for solving and factoring.

• Identify the Coefficients: In our expression, \(a = 1\), \(b = 16\), and \(c = 64\).
• Check for Special Forms: Quadratic trinomials can sometimes be perfect square trinomials or factorable by other methods.

Always try possible methods such as checking for perfect squares and other factorization techniques.

Binomial Expansion

The expansion of a binomial involves multiplying it with itself. When you have a binomial such as \((x + a)\) and you square it, the resulting expression is a perfect square trinomial, which can be expanded as follows:

• Start with \((x + a)^2\).
• Apply the distribution: multiply \((x + a)\) by \((x + a)\).
• Use the formula \((x + a)(x + a) = x^2 + 2ax + a^2\).

In the exercise, the trinomial \(x^2 + 16x + 64\) was found to be a result of expanding \((x + 8)^2\). This illustrates the binomial expansion process in obtaining a perfect square trinomial. Always verify by expanding to ensure the factorization is correct, as seen in the final verification step, where expanding \((x+8)^2\) yielded the original trinomial form, confirming the factorization was correct.