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Chapter 4: Problem 24

Find each product. $$ 2 y^{5}(y-8)(y+2) $$

Short Answer

Expert verified
The product is \(2y^7 - 12y^6 - 32y^5\).

Step by step solution

01

Distribute the polynomial

First, distribute the term outside the parentheses, 2y^5, to each term inside the parentheses \[2y^5 (y-8)(y+2) = 2y^5(y^2 + 2y - 8y - 16)\]. This effectively changes the multiplication expression from a product of three terms to a product of two terms.
02

Simplify inside parenthesis

Next, simplify the expression inside the parentheses by combining like terms \(2y^5 (y^2 - 6y - 16)\).
03

Multiply the simplified polynomial

Multiply each term inside the parentheses by 2y^5:\(2y^5 * y^2 = 2y^7\), \(2y^5 * (-6y) = -12y^6\), \(2y^5 * (-16) = -32 y^5\).
04

Combine the results

Combine the results from the multiplication step:\[2y^7 - 12y^6 - 32y^5\].

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra. It allows us to multiply a term outside the parentheses by each term inside the parentheses. This is essential when dealing with polynomial multiplication. In the exercise, we start with the expression \[2y^5 (y - 8)(y + 2)\] and use the distributive property. We first distribute \(2y^5\) to each term inside \((y - 8)\) and then multiply the result by \((y + 2)\). This turns our original expression into \[2y^5 (y^2 + 2y - 8y - 16)\] breaking it down simplifies further operations.
Combining Like Terms
Combining like terms is a key step to simplify any polynomial expression. Like terms are terms in a polynomial that have the same variables raised to the same power. In the given expression, \[2y^5 (y^2 + 2y - 8y - 16) the terms \(2y\) and \(-8y\) are like terms. So, we combine them as follows: \2y - 8y = -6y\]. Now, our expression simplifies to \[2y^5 (y^2 - 6y - 16)\]. Combining like terms makes the polynomial simpler and easier to work with, reducing errors in subsequent steps.
Exponents
Understanding exponents is crucial in polynomial multiplication. Exponents tell us how many times to multiply a number by itself. In our example, we have terms with exponents, such as \(2y^5\). When multiplying polynomials, we add the exponents of like bases. Thus in \[2y^5 * y^2\], we add the exponents of \(y\), resulting in \[2y^{5+2} = 2y^7\]. Similarly, for \[2y^5 * (-6y)\], we get \[-12y^{5+1} = -12y^6\], and finally for \[2y^5 * (-16)\], we have \[-32y^5\]. This gives us the final polynomial expression \[2y^7 - 12y^6 - 32y^5\], taking into account all exponents.

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