Problem 94 - Contemporary Precalculus

Question

(a) Let \(b\) be a real number. Multiply out the expression \(\left(x+\frac{b}{2}\right)^{2}\) (b) Explain why your computation in part (a) shows that this statement is true: If you add \(\left(\frac{b}{2}\right)^{2}\) to the expression \(x^{2}+b x,\) the resulting polynomial is a perfect square.

Step-by-Step Solution

Answer

Question: Explain why adding the term \((\frac{b}{2})^2\) to the expression \(x^2 + bx\) results in a perfect square polynomial. Answer: Adding the term \((\frac{b}{2})^2\) to the expression \(x^2 + bx\) results in a perfect square polynomial because this addition makes the expression identical to the expansion of a perfect square binomial, \((x+\frac{b}{2})^2\).

Step 1: Part (a): Expand the given expression

To expand the given expression \((x+\frac{b}{2})^2\), we need to square the binomial: \((x+\frac{b}{2})^2 = (x+\frac{b}{2})(x+\frac{b}{2})\) Now, using the distributive property (FOIL method) to multiply the terms: \((x+\frac{b}{2})(x+\frac{b}{2}) = x(x) + x\left(\frac{b}{2}\right) + \left(\frac{b}{2}\right)(x) + \left(\frac{b}{2}\right)\left(\frac{b}{2}\right)\) Simplify the expression: \(x^2 + \frac{b}{2}x + \frac{b}{2}x + \frac{b^2}{4}\) Combine the like terms: \(x^2 + bx + \frac{b^2}{4}\) So, the expanded expression is \(x^2 + bx + \frac{b^2}{4}\).

Step 2: Part (b): Explain why adding \((\frac{b}{2})^2\) results in a perfect square polynomial

Observe the expression obtained in part (a), which is \(x^2 + bx + \frac{b^2}{4}\). Notice that this is the expanded result of a perfect square of a binomial \((x+\frac{b}{2})^2\). Now, consider the given expression \(x^2 + bx\). If we add \((\frac{b}{2})^2 = \frac{b^2}{4}\) to this expression, we get: \(x^2 + bx + \frac{b^2}{4}\) We can see that this expression is the same as the expansion obtained in part (a). Therefore, by adding the term \((\frac{b}{2})^2\) to the given expression \(x^2 + bx\), the resulting polynomial is a perfect square, which is the square of the binomial \((x+\frac{b}{2})^2\).

Key Concepts

Binomial Expansion

Understanding the concept of binomial expansion is foundational when dealing with perfect square trinomials. A binomial is a simple algebraic expression containing two terms, such as \(a + b\). When we want to expand a binomial raised to a power, like \( (a + b)^2 \) for instance, we are essentially multiplying the binomial by itself.

For the given exercise, the binomial is \(x + \frac{b}{2}\). The expansion of this squared binomial gives us a trinomial, \( x^2 + bx + \frac{b^2}{4} \), which illustrates the concept that squaring a binomial will result in a polynomial with three terms. By squaring each term and then multiplying each term in the first binomial by each term in the second binomial, we obtain the coefficients that accompany each term in the resulting trinomial. It's important to realize that the middle term of a perfect square trinomial is always twice the product of the square roots of the first and last terms, as seen in the exercise with the terms \( \frac{bx}{2} + \frac{bx}{2} \).

FOIL Method

The FOIL method is a technique used for multiplying two binomials and stands for First, Outer, Inner, Last. This method is especially useful for understanding the distributive property in action. In the problem at hand, we used the FOIL method to expand \( (x + \frac{b}{2})^2 \).

Following the FOIL sequence:

• First: Multiply \( x \) by \( x \) to get \( x^2 \).
• Outer: Multiply \( x \) by \( \frac{b}{2} \) to get \( \frac{bx}{2} \).
• Inner: Multiply \( \frac{b}{2} \) by \( x \) to get another \( \frac{bx}{2} \), which is essentially the same as the outer term.
• Last: Multiply \( \frac{b}{2} \) by \( \frac{b}{2} \) to get \( \frac{b^2}{4} \).

By adding all these products together, we form the expanded binomial \( x^2 + bx + \frac{b^2}{4} \). Notice how the middle term is the sum of the outer and inner products, which is a critical step in creating the perfect square trinomial.

Distributive Property

The distributive property is a cornerstone of algebra and is used to simplify expressions. It states that multiplications distributed over addition—briefly, \( a(b + c) = ab + ac \). When we expand the binomial \( (x + \frac{b}{2})^2 \), we are essentially distributing the terms of one binomial over another.

In the exercise, applying the distributive property through the FOIL method shows us how each term in the first parenthesis is multiplied by each term in the second parenthesis. It ensures that each part of the binomial is accounted for in the final expression. The process also reveals why adding the term \( (\frac{b}{2})^2 \) to \( x^2 + bx \) makes it a perfect square: the distributive property ensures the middle term in the resulting trinomial combination is the sum of the products of the inner and outer steps, which mathematically guarantees a square when structured correctly. This explains the fundamental reason why the formula for perfect square trinomials works and is a powerful tool for verifying such expressions in algebra.