Problem 18 In Exercises 17鈥�24, fi nd the ... [FREE SOLUTION]

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BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 18

In Exercises 17鈥�24, fi nd the product. $-4 x^5\left(11 x^3+2 x^2+9 x+1\right)$

Short Answer

Expert verified

$-44 x^8 - 8 x^7 - 36 x^6 - 4 x^5$

Step by step solution

Distributing Factor

Distribute $-4 x^5$ with every term within the parenthesis. This results in: $-4 x^5 * 11 x^3, -4 x^5 * 2 x^2, -4 x^5 * 9 x, -4 x^5 * 1$

Simplifying

Simplify all the terms: \(-44 x^8,-8 x^7,-36 x^6, -4 x^5\

Writing Final Result

\-44 x^8,-8 x^7,-36 x^6, -4 x^5 are the terms resulted from the multiplication, by using commas we can group all four terms together.

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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 that makes it easier to multiply polynomials. It's quite straightforward: when you need to multiply a term by a sum of several terms inside parentheses, you distribute the multiplication over each term in the parentheses separately.
For example, using the distributive property to solve a problem like $-4x^5(11x^3 + 2x^2 + 9x + 1)$, requires multiplying $-4x^5$ by each term of the polynomial separately:

$-4x^5 \times 11x^3$
$-4x^5 \times 2x^2$
$-4x^5 \times 9x$
$-4x^5 \times 1$

This step sets the foundation for simplifying and solving polynomial expressions.

Monomials

A monomial is a single algebraic expression that can be a constant, a variable, or variables multiplied together with whole number exponents. Understanding monomials plays a crucial role when dealing with polynomial multiplication.
In our exercise, $-4x^5$ is an example of a monomial. It consists of a constant ($-4$) and a variable term ($x^5$), where $x$ is raised to the power of five.Understanding the structure of monomials is key when performing operations as it helps in simplifying and arranging terms during polynomial multiplication. Monomials are super simple to work with once you get the hang of them!

Exponents

Exponents are used to represent repeated multiplication of the same number or variable. They are integral to polynomial expressions, showing how many times a number or variable is used as a factor. For instance, $x^5$ means that $x$ is multiplied by itself five times.When multiplying monomials, exponents play a crucial role. According to the rules of exponents, when you multiply terms with the same base, you add their exponents. For example:

$-4x^5 \times 11x^3 = -44x^{5+3} = -44x^8$

This establishes how important it is to manage exponents carefully to correctly simplify expressions

Polynomial Expressions

Polynomial expressions are algebraic expressions that include constants, variables, and exponents. They can have one or more terms. A polynomial like $11x^3 + 2x^2 + 9x + 1$ contains four terms of varying degrees.In the given exercise, polynomial multiplication involves taking each term from the polynomial expression and distributing the multiplication by the monomial. After multiplying, you acquire a new polynomial, which in this case is: $-44x^8 - 8x^7 - 36x^6 - 4x^5$.Polynomials can be tricky, but breaking them down into these terms and using the steps described simplifies the process significantly.

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91影视

In Exercises 17鈥�24, fi nd the product. \(-4 x^5\left(11 x^3+2 x^2+9 x+1\right)\)

Short Answer

Step by step solution

Distributing Factor

Simplifying

Writing Final Result

Key Concepts

Distributive Property

Monomials

Exponents

Polynomial Expressions

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