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Chapter 4: Problem 43

In Exercises 43鈥�48, use Pascal鈥檚 Triangle to expand the binomial. \((2 t+4)^3\)

Short Answer

Expert verified
The expanded form of \((2t+4)^3\) using Pascal鈥檚 Triangle is \(8t^3 + 72t^2 + 192t + 64\).

Step by step solution

01

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.
02

Apply the binomial theorem to expand the expression

We have \((2t+4)^3 = 1*(2t)^3 + 3*(2t)^2*4 + 3*2t*(4)^2 + 1*(4)^3\).
03

Simplify the Result

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises 17鈥�24, fi nd the product. \((-x-3)\left(2 x^2+5 x+8\right)\)

In Exercises 25 and 26, describe and correct the error in performing the operation. \(\left(x^2-3 x+4\right)-\left(x^3+7 x-2\right)\) \(=x^2-3 x+4-x^3+7 x-2\) \(=-x^3+x^2+4 x+2\)

MODELING WITH MATHEMATICS During a recent period of time, the numbers (in thousands) of males \(M\) and females \(F\) that attend degree-granting institutions in the United States can be modeled by $$ \begin{aligned} &M=19.7 t^2+310.5 t+7539.6 \\ &F=28 t^2+368 t+10127.8 \end{aligned} $$ where \(t\) is time in years. Write a polynomial to model the total number of people attending degree- granting institutions. Interpret its constant term.

The standard equation of a circle with radius \(r\) and center \((h, k)\) is \((x-h)^2+(y-k)^2=r^2\). Rewrite each equation of a circle in standard form. Identify the center and radius of the circle. Then graph the circle. a. \(x^2+6 x+9+y^2=25\) b. \(x^2-4 x+4+y^2=9\) c. \(x^2-8 x+16+y^2+2 y+1=36\)

In Exercises 43鈥�48, use Pascal鈥檚 Triangle to expand the binomial. \((g+2)^5\)
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