Problem 49 - College Algebra with Corequisite Support

Question

Which expression cannot be expanded using the Binomial Theorem? Explain. $$\begin{array}{l} \left(x^{2}-2 x+1\right) \\ (\sqrt{a}+4 \sqrt{a}-5)^{8} \\ \left(x^{3}+2 y^{2}-z\right)^{5} \\ \left(3 x^{2}-\sqrt{2 y^{3}}\right)^{12} \end{array}$$

Step-by-Step Solution

Answer

\((x^2 - 2x + 1)\) and \((x^3 + 2y^2 - z)^5\) can't be expanded using the Binomial Theorem.

Step 1: Understanding the Binomial Theorem

The Binomial Theorem provides the expansion of expressions of the form \((a + b)^n\), where \(a\) and \(b\) are terms that can contain variables, constants, or both, and \(n\) is a non-negative integer.

Step 2: Identifying Proper Form

Each given expression must be evaluated to see if it fits the form \((a + b)^n\): exactly two terms raised to a power. Only such expressions can be expanded using the Binomial Theorem.

Step 3: Evaluate Each Expression

1. \((x^2 - 2x + 1)\) - Has three terms.2. \((\sqrt{a} + 4\sqrt{a} - 5)^8\) - Can be simplified to \((5\sqrt{a} - 5)^8\), two terms.3. \((x^3 + 2y^2 - z)^5\) - Has three terms.4. \((3x^2 - \sqrt{2y^3})^{12}\) - Has two terms.

Step 4: Identify Which Cannot Be Expanded

Expressions 1 and 3 have more than two terms, making them unsuitable for expansion via the Binomial Theorem. Only expressions that fit exactly the form \((a + b)^n\) can utilize the theorem. Therefore, \((x^2 - 2x + 1)\) and \((x^3 + 2y^2 - z)^5\) don't match the required form.

Key Concepts

Polynomials

Polynomials are mathematical expressions comprising variables and coefficients. They involve operations like addition, subtraction, and multiplication, but not division by variables. A polynomial looks like a sum of terms, where each term is a product of a constant and a non-negative integer power of a variable.

• Monomials: A single term. Example: \(3x^2\)
• Binomials: Two terms. Example: \(x^2 - 5\)
• Trinomials: Three terms. Example: \(x^2 + 3x + 2\)
• Higher degree polynomials: More than three terms. Example: \(2x^4 + 3x^3 + x^2 + 7\)

Each term in a polynomial has a coefficient and one or more variables raised to whole number exponents. Understanding these structures is crucial when dealing with expansions, especially using rules like the Binomial Theorem, which applies specifically to binomials.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division). Unlike equations, algebraic expressions do not contain an equality sign.
They include terms similar to polynomials but can be more complex, involving:

• Radicals such as \(\sqrt{x}\)
• Fractions like \(\frac{1}{x}\)
• Exponents and base variables not restricted to integers

In analyzing algebraic expressions, the Binomial Theorem provides a framework to expand powers of binomials, simplifying calculations by transforming an expression like \((a + b)^n\) into a sum of terms. Not all algebraic expressions can be expanded using this theorem, as they must specifically be in the \((a + b)^n\) format with only two distinct terms.

Expansion of Binomials

The expansion of binomials leverages the Binomial Theorem to express raised powers of binomials in an expanded form. This powerful tool is useful for simplifying expressions and solving problems efficiently in algebra.
The Binomial Theorem states that for any non-negative integer \(n\), any term of the expansion of \((a + b)^n\) is given by:\[\sum_{k=0}^n \binom{n}{k}a^{n-k}b^k\]Here, \(\binom{n}{k}\) represents the binomial coefficient, which can be calculated as \(\frac{n!}{k!(n-k)!}\), signifying the number of ways to choose \(k\) elements from \(n\) elements. This theorem transforms binomials into a sum of terms, each involving powers of the original components \(a\) and \(b\).
For a binomial to be expanded using this theorem, it must precisely fit the \((a + b)^n\) pattern. Expressions with more than two terms, such as trinomials or higher, cannot utilize the Binomial Theorem for expansion. Therefore, identifying proper binomial forms is a key step in applying this algebraic tool effectively.