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Chapter 9: Problem 9

Multiply the following expressions. $$\left(3 x^{2}-7\right) 2 x^{4}$$

Short Answer

Expert verified
The solution is \(6x^6 - 14x^4\).

Step by step solution

01

Distribute the Factor

We start by distributing the factor outside the parentheses, which is the term \(2x^4\), to each term inside the parentheses. The expression is \((3x^2 - 7) \cdot 2x^4\). Distributing gives us \(3x^2 \cdot 2x^4 - 7 \cdot 2x^4\).
02

Multiply the Terms

Multiply each term individually. For \(3x^2 \cdot 2x^4\), multiply the coefficients (3 and 2) and add the exponents of like bases \(x\), which results in \(6x^{6}\). For \(-7 \cdot 2x^4\), multiply the coefficients (-7 and 2) to get \(-14x^4\).
03

Write the Final Expression

Combine the terms obtained from multiplying: \(6x^6 - 14x^4\). This is the expanded and simplified form after multiplying the original expressions.

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Key Concepts

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

The Distributive Property in Polynomial Multiplication
The distributive property is a foundational concept in algebra that allows us to simplify multiplication across terms in an expression. When faced with an expression such as \((3x^2 - 7) \cdot 2x^4\), we use the distributive property to multiply the monomial \(2x^4\) by each term inside the parentheses.
  • This process involves taking the entire expression \(2x^4\) and applying it to each part of the binomial \((3x^2 - 7)\).
  • For the given problem, the distributive property means multiplying \(3x^2\) by \(2x^4\) and then multiplying \(-7\) by \(2x^4\).
Using this property helps in breaking down more complex multiplication problems into manageable parts, ensuring that all terms in an expression are calculated correctly.
Understanding Exponents in Multiplication
Exponents represent repeated multiplication of a base number. When multiplying terms with the same base, you add their exponents. This rule for exponents is crucial when we encounter expressions like \(3x^2 \cdot 2x^4\).
  • First, identify that both terms have the base \(x\).
  • Next, apply the rule by adding the exponents: \(2 + 4 = 6\).
As a result, \(x^2\) and \(x^4\) combine to form \(x^6\). This step leads to one part of our solution: \(6x^6\), highlighting the power of exponent addition in simplifying expressions during polynomial multiplication.
Coefficient Multiplication Explained
When dealing with polynomial multiplication, it's important to address the coefficients of each term, which are the numerical parts accompanying variables. In our example, these are the numbers we multiply directly.
  • For \(3x^2 \cdot 2x^4\), multiply the coefficients \(3\) and \(2\) to get \(6\).
  • For \(-7 \cdot 2x^4\), multiply \(-7\) and \(2\) to get \(-14\).
After multiplying coefficients, ensure you also multiply the variable parts using the exponent rules. Combining these steps results in the simplified expression, \(6x^6 - 14x^4\), demonstrating how combining coefficient multiplication with exponent rules leads to an expanded polynomial expression.

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Most popular questions from this chapter

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