Problem 41 - Algebra 2

Question

OPEN ENDED Write a power of a binomial for which the first term of the expansion is 625\(x^{4}\) .

Step-by-Step Solution

Answer

The binomial expression is \((5x + b)^4\).

Step 1: Identify the General Expression

We are given that the first term is 625\(x^4\). Recall the binomial theorem, which states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). The first term \(T_0\) of this expansion is \(a^n\). Therefore, \(a^n = 625x^4\).

Step 2: Express the First Term

From the equation \(a^n = 625x^4\), we observe that \(a\) includes the variable and its coefficient. Let's set \(a = cx^p\), so \((cx^p)^n = 625x^4\), where \(n\) is the binomial power that we need to determine.

Step 3: Determine Values of \(c\) and \(p\)

Since \((cx^p)^n = 625x^4\), by comparing terms, we get \((c^n)(x^{pn}) = 625x^4\). Equate the powers of \(x\) to get \(pn = 4\). Choose \(p = 1\) for simplicity, thus \(n = 4\). Now, \(c^n = 625\), so \(c^4 = 625\).

Step 4: Compute the Coefficient \(c\)

Solve \(c^4 = 625\). Since 625 is a perfect fourth power, \(c = 5\). So the coefficient \(c\) is 5, and as per our choice \(n = 4\), \(a = 5x\).

Step 5: Find the Binomial Expression

Now that we have determined the form of \(a = 5x\) and the power \(n = 4\), the full expression becomes \((5x + b)^4\) where the first term expands to \(625x^4\) as required.

Key Concepts

Binomial Expansion

The binomial expansion is an essential concept within algebra that allows us to expand expressions raised to a power. The classic binomial theorem provides a way to expand expressions of the form \((a + b)^n\). It involves summing terms that consist of products of combinations and powers of \(a\) and \(b\). Specifically:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Each term in this expansion is a combination of integer coefficients, and powers of \(a\) and \(b\).

This method is invaluable for simplifying algebraic expressions, especially when tackling complex algebra problems or calculus derivatives.

Powers of Binomials

When dealing with powers of binomials, each binomial is raised to a specific power, denoted by \(n\). A binomial is simply an algebraic expression containing two terms. For example, \((a + b)^4\) is a power of a binomial.

• The power \(n\) dictates how many terms are in the expansion.
• The first term in the expansion comes from raising the first term of the binomial to the power of \(n\).

Recognizing how the powers of \(a\) and \(b\) decrease and increase, respectively, as we move through the expanded terms, helps in easily predicting or identifying terms within the expansion.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations. They form the backbone of algebra and allow us to express mathematical problems and their solutions. In the context of our exercise:

• We assigned an expression form \(a = cx^p\) in the first term \(625x^4\), which simplifies solving for binomial components.
• Identifying the coefficient \(c\) and the power \(p\) of the variable is key to solving the expression correctly.

Grasping the structure of algebraic expressions aids in understanding more complex expressions, facilitating manipulations such as expansion, simplification, and factorization.