Problem 44 Find each product. $$ (y+2)^... [FREE SOLUTION]

Chapter 4: Problem 44

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

Short Answer

Expert verified

The product is $ y^3 + 6y^2 + 12y + 8 $.

Step by step solution

- Understand the Binomial Theorem

The binomial theorem states that \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \] In this case, $a = y$, $b = 2$, and $n = 3$.

- Write the Binomial Expansion

Apply the binomial theorem for $(y+2)^3$: \[ (y+2)^3 = \binom{3}{0}y^{3-0}2^0 + \binom{3}{1}y^{3-1}2^1 + \binom{3}{2}y^{3-2}2^2 + \binom{3}{3}y^{3-3}2^3 \]

- Calculate Binomial Coefficients

Calculate the binomial coefficients: $ \binom{3}{0} = 1 $, $ \binom{3}{1} = 3 $, $ \binom{3}{2} = 3 $, and $ \binom{3}{3} = 1 $.

- Substitute and Simplify Each Term

Substitute the binomial coefficients and simplify each term: \[ (y+2)^3 = 1 \cdot y^3 \cdot 1 + 3 \cdot y^2 \cdot 2 + 3 \cdot y \cdot 4 + 1 \cdot 1 \cdot 8 \]This simplifies to \[ y^3 + 6y^2 + 12y + 8 \]

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Key Concepts

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

Binomial Expansion

The concept of binomial expansion allows us to expand expressions of the form $ (a + b)^n $ into a series. This is extremely useful when dealing with polynomials. The binomial theorem provides a systematic method to expand these expressions. According to the binomial theorem, $ (a + b)^n $ can be represented as the sum of terms involving binomial coefficients, powers of $a $, and powers of $b $. For example, to expand $ (y + 2)^3 $, we use the binomial theorem to find: \[ (y + 2)^3 = \binom{3}{0}y^{3}2^0 + \binom{3}{1}y^{2}2^1 + \binom{3}{2}y^{1}2^2 + \binom{3}{3}y^{0}2^3 \] This forms a series of terms that we can further simplify.

Polynomial Multiplication

Polynomial multiplication is the process of expanding expressions like $ (y + 2)^3 $. Each term in the polynomial is multiplied by every term in the other polynomial. However, using the binomial theorem, it becomes easier to systematically multiply and expand these terms. The expanded form of $ (y + 2)^3 $ after applying binomial expansion includes: \[ y^3 + 6y^2 + 12y + 8 \] Here, each coefficient is derived from the binomial coefficient phenomenon and each term represents the multiplied and simplified powers of $y $ and $2 $.

Binomial Coefficients

The values $ \binom{n}{k} $ are known as binomial coefficients, and they are a crucial part of the binomial theorem. These coefficients can be found using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] For the example $ (y + 2)^3 $, the binomial coefficients are: \[ \binom{3}{0} = 1, \binom{3}{1} = 3, \binom{3}{2} = 3, \text{ and } \binom{3}{3} = 1 \] These coefficients indicate how many ways you can choose positions to insert each term during multiplication. They help us form the expanded series, allowing for a simpler and more organized polynomial solution. Thus, binomial coefficients are a fundamental tool for working with and expanding binomials.

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91影视

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

Short Answer

Step by step solution

- Understand the Binomial Theorem

- Write the Binomial Expansion

- Calculate Binomial Coefficients

- Substitute and Simplify Each Term

Key Concepts

Binomial Expansion

Polynomial Multiplication

Binomial Coefficients

One App. One Place for Learning.

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