Problem 101 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(4 w+1)^{2}$$

Step-by-Step Solution

Answer

The special product of the given binomial squared \((4w + 1)^2\) is \(16w^2 + 8w + 1\).

Step 1: Identify the terms a and b in the given binomial

In the given binomial \((4w + 1)^2\), we have \(a = 4w\) and \(b = 1\).

Step 2: Apply the square of a binomial formula

The general formula for the square of a binomial is: \((a + b)^2 = a^2 + 2ab + b^2\). Now, let's substitute the values of \(a\) and \(b\) from the given binomial: \((4w + 1)^2 = (4w)^2 + 2(4w)(1) + (1)^2\)

Step 3: Simplify the expression

Now, we can simplify the expression to get our final answer: \((4w + 1)^2 = 16w^2 + 8w + 1\) The special product of the given binomial squared is \(16w^2 + 8w + 1\).

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential concept in algebra. It provides a formula to expand expressions that are raised to a power. If you have an expression in the form \((a + b)^n\), the theorem allows you to express it as a series of terms. Each term is a product of a binomial coefficient and powers of both a and b.

Here's a simple representation of the binomial theorem for expanding powers:

• The expression \((a + b)^2\) expands to \(a^2 + 2ab + b^2\).
• In general terms, each term in the expansion is \( \binom{n}{k}a^{n-k}b^k \), where \( \binom{n}{k} \) is the binomial coefficient.

Applying this theorem to specific problems, such as the one described above with \((4w + 1)^2\), makes it much easier to break down and calculate expansions of higher degree polynomials. It's incredibly useful for both theoretical and applied mathematics.

Polynomial Expansion

Polynomial Expansion refers to the process of expressing a polynomial in its extended form. To expand a polynomial like \((a + b)^2\), you apply the rules of multiplication and the binomial theorem. Each term in the expanded form is created by multiplying the terms of the original polynomial.

In the example provided where \((4w + 1)^2\) was expanded, the aim is to express the polynomial in its fullest extent without using exponents repeatedly. Here’s how it works:

• First, identify the terms to be expanded: in this case, they are \(4w\) and \(1\).
• Next, apply the formula: \(a^2 + 2ab + b^2\).
• Finally, simplify to obtain the polynomial: \(16w^2 + 8w + 1\).

Expanding polynomials helps in simplifying algebraic expressions and is a fundamental step in solving equations, analyzing functions, and optimizing problems.

Special Products

Special Products in algebra include specific patterns of multiplication that simplify calculations. Recognizing these patterns can save time and effort in solving equations. Common examples include the square of a binomial: \((a + b)^2\), the difference of squares, and the product of a sum and difference.

For the exercise \((4w + 1)^2\), it represents a classic case of the square of a binomial that can be expanded using its known formula: \((a + b)^2 = a^2 + 2ab + b^2\). This formula transforms our example into \(16w^2 + 8w + 1\).

• The square of binomials: \((a + b)^2\) and \((a - b)^2\) yield well-known patterns.
• The difference of two squares: \(a^2 - b^2\) simplifies to \((a + b)(a - b)\).
• These products aid in fast tracking calculations.

Understanding special products is crucial to mastering polynomial operations and forms the basis for more advanced algebraic concepts.