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Chapter 10: Problem 17

Multiply. $$8 w\left(-3 w^{2}-2 w+4\right)$$

Short Answer

Expert verified
-24w^{3}-16w^{2}+32w

Step by step solution

01

- Distribute the 8w to each term

Apply the distributive property by multiplying 8w with each term inside the parentheses \[-3w^2\]-2w+4): 8w(-3w^2)+8w(-2w)+8w(4)
02

- Simplify each multiplication

Simplify each product separately: - ( 8w (-3w^{2})=8 - - 3 - w ^{2 w } = 8 \times -3= - 24 \times w^3 = \[-3w^3\] - 8w( - 2 ) \8-2w^2 balance w^2 = - -2 8w so -16w^2 =-16 a 8w(4)=8*4=32=a8w4
03

- Combine results

Combine the results from Step 2: -24w^{3}-16w^{2}+32w

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Key Concepts

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

Multiplication of Polynomials
Multiplying polynomials involves distributing each term of one polynomial to every term of the other polynomial. This might seem complex at first, but practicing the distributive property makes it easier.

Consider the polynomial multiplication: 8w(-3w^2-2w+4). The steps are:

First, distribute 8w to each term inside the parentheses:
  • 8w * -3w^2
  • 8w * -2w
  • 8w * 4
Next, simplify each resulting multiplication:

  • 8w * -3w^2 = -24w^3
  • 8w * -2w = -16w^2
  • 8w * 4 = 32w
Finally, combine these results to get the final product: -24w^3 - 16w^2 + 32w.This is the result of multiplying the given polynomials.
Distributive Property
The distributive property is a key principle in algebra that simplifies expressions and makes problem-solving more manageable. It states that for any numbers or expressions a, b, and c, the following holds true:

a(b + c) = ab + ac

In the exercise: 8w(-3w^2 - 2w + 4), we use the distributive property by multiplying 8w with each term inside the parentheses. Let's break it down:
  • 8w * (-3w^2) = -24w^3
  • 8w * (-2w) = -16w^2
  • 8w * (4) = 32w
Using the distributive property allows us to handle each term individually, simplifying the process. It turns complex expressions into manageable steps, and it is essential for solving algebraic equations.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves combining like terms and reducing the expression to its simplest form. This allows for clearer and more manageable equations. Let's revisit our final combined terms from the exercise:

-24w^3 - 16w^2 + 32w.

Each term in this expression is different because they have different powers of w (w^3, w^2, and w). Therefore, no further simplification from combining like terms is possible.

Simplifying is crucial in algebra, not just for polynomial multiplication but for any expression. It helps to:

  • Ensure the expression is as uncomplicated as possible.
  • Make further calculations or solving equations more manageable.
  • Provide a clear understanding of the problem.
Always check your final result to ensure it is simplified correctly. This enhances clarity and accuracy in mathematical expressions.

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Most popular questions from this chapter

Simplify the expression. $$\left(4 x y^{3}\right)^{3}$$

Perform the indicated operations. $$\left(-9 m^{4}-7 m^{2}+8\right)-\left(3 m^{4}+9 m^{3}-11\right)-\left(8 m^{3}+m^{2}-3\right)$$

Simplify the expression. $$\left(2 x y^{4}\right)^{3}\left(4 x^{2} y^{3}\right)$$

Simplify the expression. $$(3 t)^{2}$$

Simplify. Write the answers with positive exponents only. $$\left(w^{-7}\right)^{2}\left(w^{3}\right)^{4}$$
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