Problem 107 - Algebra 2

Question

Expand each binomial. $$ \left(x^{2}+x\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form of the binomial \( (x^{2}+x)^4 \) is \( x^{8} + 4x^{7} + 6x^{6} + 4x^{5} +x^{4} \).

Step 1: Identify the binomial and the power

The binomial is \( (x^{2}+x) \) and it is raised to the power 4.

Step 2: Apply Binomial Theorem for the expansion

We apply the Binomial Theorem to expand the binomial. This theorem can be written in this form: \( (a+b)^n = Σ (nCk) * a^(n-k) * b^k \) where the summation Σ runs from k=0 to n.

Step 3: Replace a,b,n in Binomial Theorem

We replace a by \( x^{2} \), b by x and n by 4 in Binomial Theorem. This will give us: \( (x^{2}+x)^4 = Σ (4Ck) * (x^{2})^(4-k) * x^k \).

Step 4: Calculate the values for each term

We calculate every term in the summation for k ranging from 0 to 4. Each term can be represented as: Term_k = \(4Ck * (x^{2})^{4-k} * x^k \). We substitute k into this equation to find each term. This gives us:Term_0 = \( 4C0 * (x^{2})^4 * x^0 = 1 * x^{8} = x^{8} \)Term_1 = \( 4C1 * (x^{2})^3 * x^1 = 4 * x^{6} * x= 4x^{7} \)Term_2 = \( 4C2 * (x^{2})^2 * x^2 = 6 * x^{4} * x^{2} = 6x^{6} \)Term_3 = \( 4C3 * (x^{2})^1 * x^3 = 4 * x^{2} * x^{3} = 4x^{5} \)Term_4 = \( 4C4 * (x^{2})^0 * x^4 = 1 * x^{4} = x^{4} \)

Step 5: Add together all terms

Now we add together the terms calculated in the previous step to give the final expanded binomial: \( x^{8} + 4x^{7} + 6x^{6} + 4x^{5} +x^{4} \).

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that helps us expand expressions raised to any positive integer power. It's particularly useful for simplifying problems involving powers of binomials, which are expressions of the form \((a+b)^n\). This theorem expresses the expanded form as a series of terms using combinations and powers.

The theorem provides a formula for writing the expression more concretely:

• Each term involves a combination denoted by \(nCk\) or \(\binom{n}{k}\), which indicates the number of ways to choose \(k\) elements from \(n\) elements without regard to order.
• The expression \(a^{(n-k)}\) represents the decreasing powers of the first term "a."
• The expression \(b^k\) represents the increasing powers of the second term "b."

When expanding \((x^2 + x)^4\) using this theorem, we substitute \(a = x^2\), \(b = x\), and \(n = 4\). This guides us through calculating the powers and combinations for each term up to \(n\), leading to a complete polynomial expansion.

Polynomial Expansion

Polynomial expansion involves rewriting a binomial raised to an exponent as a sum of multiple terms. Each term is a product of coefficients derived from the binomial coefficients and the powers of the terms involved in the binomial.

In our exercise, we are given \((x^2 + x)^4\) which requires expansion into a polynomial. Applying the Binomial Theorem, we calculate individual terms by:

• Determining the coefficients using combinations, such as \(4C0, 4C1, 4C2\), etc.
• Calculating the powers of the terms: \((x^2)^{4-k}\) and \(x^k\).

For example, a term with \(k=0\) would be \(4C0 \times (x^2)^4 \times x^0 = x^8\), contributing to the polynomial structure of the answer.

Each step builds the polynomial by increasing the number of terms, allowing us to see how the original binomial transforms into a complete expression composed of different power terms. The final result of \(x^8 + 4x^7 + 6x^6 + 4x^5 + x^4\) showcases how the powers distribute and combine neatly into a single expanded form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that combines numbers, variables, and operators. They form the basis of algebra and are utilized to express complex mathematical concepts in simpler, structured forms.

In algebra, binomials are one type of algebraic expression consisting of two terms, such as \((x^2 + x)\). Expanding this binomial using the Binomial Theorem turns it into a more comprehensive polynomial expression.

• Each term in the expanded polynomial retains the core components of the original expression, like exponents and coefficients.
• The process illustrates how algebraic techniques can simplify and distribute complex operations.

Handling algebraic expressions correctly is crucial as it forms the foundation for solving equations and modeling real-world situations using mathematics.

Understanding how to work with them, like expanding a binomial, enables students to navigate more sophisticated algebraic problems with ease, honing skills that are essential in higher-level mathematics and various applied fields.