Question

In Exercises 43–48, use Pascal’s Triangle to expand the binomial. \((2 t+4)^3\)

Short answer

The expanded form of \((2t+4)^3\) using Pascal’s Triangle is \(8t^3 + 72t^2 + 192t + 64\).

Step-by-step solution

Identify the coefficients from Pascal's Triangle

Look at the 3rd row of Pascal's Triangle (since we're considering a cube, or \(x^3\)). The coefficients for the expansion in this case are 1, 3, 3, 1.

Apply the binomial theorem to expand the expression

We have \((2t+4)^3 = 1*(2t)^3 + 3*(2t)^2*4 + 3*2t*(4)^2 + 1*(4)^3\).

Simplify the Result

Carry out the multiplications to get the final expanded form, which is \(8t^3 + 72t^2 + 192t + 64\).

Key concepts

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers that gives us coefficients for expanding binomials. Each row corresponds to the powers of a binomial expansion. For example, the third row

• 1
• 3
• 3
• 1

represents the coefficients needed for expanding \((a + b)^3\). To build Pascal's Triangle, start with a "1" at the top. Each subsequent row starts and ends with "1," with each interior number being the sum of the two numbers directly above it in the previous row. This progression makes it a handy tool for quick reference when dealing with powers in binomials.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions that are raised to any power. It is represented as:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(\binom{n}{k}\) are the binomial coefficients from Pascal's Triangle. This theorem allows us to break down expressions like \((2t + 4)^3\) into manageable parts. By applying the coefficients we derived from Pascal's Triangle:

• Multiply each term in the binomial by the corresponding coefficient.
• Raise each part to the decreasing and increasing powers of the binomial terms.

This step-by-step breakdown simplifies complex algebraic expressions effectively.

Polynomial Expansion

Polynomial expansion involves rewriting a binomial raised to a power as a sum of its components. Through expanding \((2t + 4)^3\), we translate it into a polynomial made up of various terms. This process:

• Transforms the original compact expression into a series of terms with coefficients, variables, and constants.
• Breaks down each multiplication to simplify solving and further operations.

As we expand, ensure each term is carefully calculated by maintaining correct multiplication and alignment with the binomial coefficients. This precision transforms an expression into its detailed expanded form, \(8t^3 + 72t^2 + 192t + 64\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and their operations, like addition and multiplication. In the binomial expansion of\((2t + 4)^3\),we manage complex algebraic expressions to form a polynomial. Consider:

• Each term represents a part of the original expression's expansion.
• Understanding how to simplify these expressions is crucial for solving equations and inequalities.

Simplifying these expressions involves combining like terms and ensuring multiplication is accurately executed. This comprehension of algebraic expressions is foundational in algebra, as it assists in further explorations and calculations.