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Chapter 3: Problem 24

For Problems \(1-36\), find each product. $$ \left(-7 x^{2}\right)(3 x)\left(4 x^{3}\right) $$

Short Answer

Expert verified
\(-84x^6\)

Step by step solution

01

Identify and Group Coefficients

Identify the coefficients in each term. The expression \((-7x^2)(3x)(4x^3)\) has coefficients -7, 3, and 4. We will multiply these coefficients together: \(-7 \times 3 \times 4\).
02

Multiply Coefficients

Calculate the product of the coefficients: -7 multiplied by 3 is -21, and then -21 multiplied by 4 gives -84. So, the product of the coefficients is -84.
03

Identify and Group the Variables

Each term contains the variable \(x\) raised to some power: \(x^2\), \(x^1\), and \(x^3\). We need to multiply these variable terms together.
04

Add the Exponents

When multiplying variables with the same base, add their exponents. So, add the exponents: 2 (from \(x^2\)) + 1 (from \(x\)) + 3 (from \(x^3\)) to get the exponent of the combined variable term: \(2 + 1 + 3 = 6\).
05

Combine the Products

Combine the coefficient product and the variable product: The coefficient product is -84 and the variable term is \(x^6\). So, the final product is \(-84x^6\).

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Key Concepts

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

Coefficients in Polynomial Multiplication
Coefficients are the numerical factors in terms of a polynomial expression. In simple terms, they are the numbers that sit in front of the variables. Understanding how to use them is crucial when it comes to polynomial multiplication.
  • In the expression \((-7x^2)(3x)(4x^3)\), the coefficients are -7, 3, and 4.
  • To solve for the product, we first focus on multiplying these coefficients together.
  • Start by multiplying -7 and 3 to get -21. Then multiply -21 by 4 to arrive at -84.
The correct handling of coefficients ensures accurate calculation of the numerical part of the product. Practicing with coefficients separately can help solidify this foundational aspect of algebra.
Understanding Exponents
Exponents represent the number of times a base is multiplied by itself. They are a key component in algebra, especially when dealing with terms that require multiplication.
For instance, in our example, we have the expression \((-7x^2)(3x)(4x^3)\). Here, the variable \(x\) is raised to different powers, specifically exponent 2 in \(x^2\), exponent 1 in \(x\), and exponent 3 in \(x^3\).
  • When multiplying terms with the same base, the exponents are added together.
  • In this case, you add 2, 1, and 3 to get 6, resulting in \(x^6\).
Adding exponents correctly is vital to finding the right product of powers when multiplying like terms. Using this method simplifies the process, avoiding errors in complex calculations.
Algebraic Expressions
Algebraic expressions consist of sums and products involving variables and coefficients. When multiplying expressions, knowing how to manage these components is key to simplification.
The expression \((-7x^2)(3x)(4x^3)\) combines numeric coefficients and variables with exponents. Processing these effectively requires a step-by-step approach.
  • The first step is to isolate and multiply each element, starting with coefficients and following through with variables.
  • Once individual products are calculated, they are assembled into a single expression, like \(-84x^6\).
By understanding each part of an algebraic expression and how to manipulate them, students can solve polynomial multiplication problems with confidence. Practicing these steps enhances problem-solving skills, increasing proficiency in algebra overall.

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Most popular questions from this chapter

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ n^{2}+20 n+91=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ -3 x^{2}-19 x+14=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ (x+8)(x-6)=-24 $$

For Problems 104-109, factor each trinomial and assume that all variables that appear as exponents represent positive integers. $$ 12 x^{2 n}+7 x^{n}-12 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ x^{2}+4 x+3=0 $$
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