Problem 73 Multiply: $$\left(x^{2}+2 x y+... [FREE SOLUTION]

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Chapter 14: Problem 73

Multiply: $$\left(x^{2}+2 x y+y^{2}\right)\left(x^{2}+2 x y+y^{2}\right)^{2}(x+y).$$

Short Answer

Expert verified

(x + y)^5.

Step by step solution

Identify the Expression

The given expression is $(x^{2} + 2xy + y^{2})(x^{2} + 2xy + y^{2})^{2}(x + y)$.

Simplify the Squared Term

Recall that the squared term $(x^{2} + 2xy + y^{2})$ can be rewritten as $(x + y)^2$. Therefore, the expression simplifies to $(x + y)^{2}(x + y)^{2}(x + y)$.

Combine Like Bases

When multiplying terms with the same base, add the exponents. So, $(x + y)^{2} \times (x + y)^{2} \times (x + y) = (x + y)^{2+2+1} = (x + y)^5$.

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Key Concepts

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

Exponent Rules

Exponent rules are foundational for dealing with polynomial multiplication and other algebraic operations. When multiplying expressions with the same base, you simply add the exponents. For example, $(x^a) \times (x^b) = x^{a+b}$. Similarly, if you have $(x^a)^b$, it equals $(x^{a \times b})$. These rules help you simplify complex polynomial expressions by breaking them down into manageable parts. They are essential when dealing with terms like $(x + y)^2$ because they make it easier to manipulate and simplify the expressions.

Polynomial Expressions

Polynomial expressions consist of terms in the form $(ax^n)$, where $a$ is the coefficient and $n$ is the exponent. These terms can be added, subtracted, and multiplied. For example, in our given problem, the expression $(x^2 + 2xy + y^2)$ should be recognized as a polynomial. Polynomials follow specific rules during manipulation, especially when being multiplied. For instance, recognizing that $(x^2 + 2xy + y^2) = (x + y)^2$ can simplify later steps. These simplifications make polynomial multiplication more manageable and highlight the importance of understanding polynomial structure.

Simplifying Expressions

Simplifying expressions involves breaking down complex terms into simpler ones for easier computation. In our given problem, we simplified $(x^2 + 2xy + y^2)$ into $(x + y)^2$, which then allowed us to multiply $(x + y)^2$ with another $(x + y)^2$ and an $(x + y)$. This process involves combining like bases and adding their exponents: $(x + y)^2 \times (x + y)^2 \times (x + y) = (x + y)^{2+2+1} = (x + y)^5$. Breaking down and recombining expressions like this is crucial in simplifying and solving polynomial multiplication problems.

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91影视

Multiply: $$\left(x^{2}+2 x y+y^{2}\right)\left(x^{2}+2 x y+y^{2}\right)^{2}(x+y).$$

Short Answer

Step by step solution

Identify the Expression

Simplify the Squared Term

Combine Like Bases

Key Concepts

Exponent Rules

Polynomial Expressions

Simplifying Expressions

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