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Chapter 5: Problem 44

Find each product. \((y+2)^{3}\)

Short Answer

Expert verified
\( (y + 2)^{3} = y^{3} + 6y^{2} + 12y + 8 \)

Step by step solution

01

Recognize the formula

The given expression \((y + 2)^{3}\) can be expanded using the binomial theorem \( (a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3} \). In this case, \(a = y\) and \(b = 2\).
02

Apply the binomial theorem

Substitute \(a = y\) and \(b = 2\) into the formula: \( (y + 2)^{3} = y^{3} + 3(y^{2})(2) + 3(y)(2^{2}) + 2^{3} \).
03

Simplify the terms

Simplify each term: \(y^{3} + 3(y^{2})(2) + 3(y)(2^{2}) + 2^{3} = y^{3} + 6y^{2} + 12y + 8 \).
04

Write the final expression

Combine the simplified terms to get the final expanded form: \[ (y + 2)^{3} = y^{3} + 6y^{2} + 12y + 8 \].

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Key Concepts

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

Polynomial Expansion
Polynomial expansion involves breaking down expressions like \( (y+2)^{3} \). It uses the binomial theorem to handle expressions raised to a power. The binomial theorem states that \( (a+b)^{n} \) can be expanded into a sum involving terms of the form \( a^{k}b^{n-k} \) with coefficients derived from the binomial coefficients. This theorem makes it easier to expand polynomials without having to multiply everything out manually. This is important because it simplifies complex expressions and gives a clear, simplified form that is easier to work with in algebra.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. In \( (y+2)^{3} \), \ y \ and 2 are parts of the expression. Algebraic expressions are fundamental because they form the basis for building equations and functions. These expressions allow us to describe relationships and patterns in mathematics. For example, the expression \(y^{3} + 6y^{2} + 12y + 8 \), obtained from expanding \( (y+2)^{3} \), shows how the initial simple expression evolves into a complex polynomial.
Exponents and Powers
Exponents and powers are mathematical operations that involve numbers being raised to a certain power. In the expression \( (y+2)^{3} \), the 3 is the exponent, and it indicates how many times \( y+2 \) is multiplied by itself. Understanding exponents is crucial because they are used in many areas of mathematics, including polynomial expansion and solving algebraic equations. Exponents simplify writing multiplied terms and help in performing operations like differentiating and integrating functions in calculus.

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Most popular questions from this chapter

Evaluate. $$ 10,000(36.94) $$

Use scientific notation to calculate the answer to each problem. On February \(17,2009,\) Congress raised the U.S. government's debt limit to \(\$ 1.2 \times 10^{13} .\) When this national debt limit is reached, about how much will it be for every man, women, and child in the country? Use 300 million as the population of the United States.

Perform each division. $$ \frac{4 t^{2}+t^{4}+7}{t^{2}+1} $$

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 301 \times 299 $$

Graph each equation by completing the table of values. \(y=x^{2}-4\) \(\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {} \\ \hline-1 & {} \\\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline\end{array}\)
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