Problem 17 - Algebra 1: Concepts and Skills

Question

Tell whether the expression is the square of a binomial. $$ a^{2}+8 a+16 $$

Step-by-Step Solution

Answer

Yes, the given expression \( a^{2}+8 a+16 \) is a square of a binomial. In fact, it is equal to \( (a+4)^2 \).

Step 1: Identify the Coefficients

Comparing the given expression \( a^{2} + 8a + 16 \) to the general binomial square structure \( (x + y)^{2} = x^2 + 2xy + y^2 \), we can identify that \( x = a \), \( 2xy = 8a \), and \( y^2 = 16 \).

Step 2: Determine the Value of y

From the equation \( 2xy = 8a \), we can solve for \( y \) by dividing both sides by \( 2a \), which yields \( y = 4 \).

Step 3: Check the Square Value

To ensure our solution is correct, we check if \( y^2 \) from the general form matches the last term in our given expression. \( y^2 = 16 \), which indeed matches the last term in the given expression \( a^{2}+8 a+16 \).

Step 4: Confirm the Binomial Square

The given expression can now be rewritten in binomial square form as \( (a + 4)^2 \) as all coefficients match the conditions of a binomial square.

Key Concepts

Binomial Expansion

The term "binomial expansion" refers to the process of expanding an expression that is raised to a power. A binomial expression is one that contains two terms, such as \((x + y)\). When you square a binomial expression, you use the formula \((x + y)^2 = x^2 + 2xy + y^2\). This formula can be used to expand the squared binomial into a polynomial with three terms.In binomial expansion, understanding the structure makes it easier to recognize specific patterns in expressions. In our example, the expression \(a^2 + 8a + 16\) fits perfectly into the binomial expansion framework. We can identify the structure as \((x + y)^2\) where \(x = a\) and \(y = 4\). Knowing this formula helps us quickly rewrite the expanded expression as the square of a binomial.

Factoring Polynomials

Factoring is the process of breaking down a complex polynomial into simpler terms, or factors, that when multiplied together give back the original polynomial. By factoring, we can identify relationships between the polynomial's terms.Take the polynomial \(a^2 + 8a + 16\) from our example. By recognizing this as a perfect square trinomial, it can be factored into its base components.

• First, notice that the square root of the first term, \(a^2\), is \(a\).
• Second, ascertain the square root of the last term, \(16\), which is \(4\).
• Lastly, ensure that the middle term, \(8a\), satisfies the condition \(2xy\) where \(x = a\) and \(y = 4\).

Thus, the polynomial can be rewritten and factored as \((a + 4)^2\), confirming our understanding of polynomial factoring as applied to binomials.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest power of the variable is 2, typically written in the standard form \(ax^2 + bx + c\). These expressions appear frequently in algebra and represent a wide range of parabolic curves.In the example \(a^2 + 8a + 16\), we observe it is a quadratic expression with \(a = 1\), \(b = 8\), and \(c = 16\). These coefficients are crucial in understanding the structure and behavior of the quadratic, allowing us to determine its roots and vertex.Quadratic expressions can often be rewritten as the square of a binomial. This transformation simplifies solving equations and understanding the graph of the quadratic equation. By identifying it as a perfect square trinomial, we can express \(a^2 + 8a + 16\) as \((a + 4)^2\), making it easier to handle and interpret in mathematical problems.