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Chapter 4: Problem 58

Find each product. $$ (p+3)^{3} $$

Short Answer

Expert verified
(p+3)^3 = p^3 + 9p^2 + 27p + 27

Step by step solution

01

Understand the Formula

Recognize that \( (p+3)^3 \) is a binomial expression raised to a power. This can be expanded using the Binomial Theorem.
02

Binomial Theorem

The Binomial Theorem states: \[ (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \] Here, \( a = p \), \( b = 3 \), and \( n = 3 \).
03

Identify Terms

Write out the general form for \( (p+3)^3 \): \[ (p+3)^3 = {3 \choose 0} p^3 (3)^0 + {3 \choose 1} p^2 (3)^1 + {3 \choose 2} p^1 (3)^2 + {3 \choose 3} p^0 (3)^3 \]
04

Calculate Binomial Coefficients

Calculate each binomial coefficient: \[ {3 \choose 0} = 1, \quad {3 \choose 1} = 3, \quad {3 \choose 2} = 3, \quad {3 \choose 3} = 1 \]
05

Expand Each Term

Combine the coefficients and the powers: \[ 1 \cdot p^3 \cdot 1 + 3 \cdot p^2 \cdot 3 + 3 \cdot p \cdot 9 + 1 \cdot 1 \cdot 27 \]
06

Simplify Expression

Simplify the binomial expression: \[ p^3 + 9p^2 + 27p + 27 \]

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Key Concepts

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

Understanding Binomial Expansion
The binomial expansion is a way to express the power of a binomial as a sum of terms involving powers of the binomial's components and binomial coefficients.

For example, in the expression \(\( (p+3)^3 \)\), we use the binomial theorem to expand it. The binomial theorem tells us that any binomial raised to a power \(n\) can be expanded as follows:
\[ (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \].

Here, \((a + b)\) is the binomial and \(n\) is the power we are raising it to. By identifying the components of our specific binomial, where in our case \((a = p)\) and \((b = 3)\), and \(n=3\), we can expand \( (p + 3)^3 \) as shown in the steps.
Introduction to Binomial Coefficients
Binomial coefficients are the numerical factors that multiply the terms in the expansion of a binomial expression. They are represented as \({n \choose k}\), and calculated using the formula:
\[ {n \choose k} = \frac{n!}{k!(n-k)!} \].

In our example \((p + 3)^3 \), the coefficients are calculated for \(n = 3\) and \(k = 0, 1, 2, 3\). Let's determine them:
  • For \( k = 0 \): \[ {3 \choose 0} = \frac{3!}{0!(3-0)!} = 1 \].

  • For \( k = 1 \): \[ {3 \choose 1} = \frac{3!}{1!(3-1)!} = 3 \].

  • For \( k = 2 \): \[ {3 \choose 2} = \frac{3!}{2!(3-2)!} = 3 \].

  • For \( k = 3 \): \[ {3 \choose 3} = \frac{3!}{3!0!} = 1 \].
These coefficients help specify the weight of each term in the expansion.
Evaluating Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). When we expand a binomial expression, we break it down into simpler algebraic expressions.

For \( (p + 3)^3 \), the expanded form includes several algebraic expressions added together:
\[ p^3 + 9p^2 + 27p + 27 \]

Each term is derived by multiplying the binomial coefficients with the respective powers of the variables. In detail:
  • \(1 \cdot p^3 \cdot 1 = p^3 \).

  • \(3 \cdot p^2 \cdot 3 = 9p^2 \).

  • \(3 \cdot p \cdot 9 = 27p \).

  • \(1 \cdot 1 \cdot 27 = 27 \).
Adding these algebraic expressions gives the simplified expanded form.

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Most popular questions from this chapter

Find each product. $$ -8 r^{3}\left(5 r^{2}+2\right)\left(5 r^{2}-2\right) $$

Perform each indicated operation. \(\left(x^{2}+x\right)-\left(3 x^{2}+2 x-1\right)\)

If an object is projected upward under certain conditions, its height in feet is given by the trinomial $$-16 t^{2}+60 t+80$$ where \(t\) is in seconds. Evaluate this trinomial for \(t=2.5 .\) Use the result to fill in the blanks: If ________ seconds have elapsed, then the height of the object is ____________ feet.

Perform each indicated operation. \(\left[\left(3 x^{2}-2 x+7\right)-\left(4 x^{2}+2 x-3\right)\right]-\left[\left(9 x^{2}+4 x-6\right)+\left(-4 x^{2}+4 x+4\right)\right]\)

Add or subtract as indicated. \((6 b+3 c)+(-2 b-8 c)\)
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