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Chapter 9: Problem 5

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

Short Answer

Expert verified
Multiply and distribute: \(6x^6 - 14x^4\).

Step by step solution

01

Understand the Distribution

Before multiplying, recognize that you need to distribute the term outside the parentheses, which is \(2x^4\), to each term inside the parentheses \((3x^2 - 7)\). This involves two separate multiplications.
02

Distribute to the First Term

Multiply \(2x^4\) by the first term, \(3x^2\). When multiplying powers of \(x\), add their exponents: \[2x^4 imes 3x^2 = 2 imes 3 imes x^{4+2} = 6x^6\]
03

Distribute to the Second Term

Now multiply \(2x^4\) by the second term, \(-7\). Keep the negative sign in mind: \[2x^4 imes -7 = -7 imes 2 imes x^4 = -14x^4\]
04

Combine the Results

Add the two results from Step 2 and Step 3 together: \[6x^6 - 14x^4\] This is the simplified expression after distribution.

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Key Concepts

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

Distribution Property
The Distribution Property is a fundamental concept in algebra. It helps break down expressions to make them easier to solve, especially during polynomial multiplication. When you see an expression like \(2x^4(3x^2 - 7)\), you're going to distribute the term outside the parentheses across each term inside the parentheses. This means:
  • Multiply \(2x^4\) by \(3x^2\)
  • Multiply \(2x^4\) by \(-7\)
Each term gets its turn with the term that's outside. In essence, distribution spreads the multiplication over all the terms inside a parenthesis, ensuring each term is multiplied thoroughly. This property is essential for tackling and simplifying algebraic expressions easily.
Exponent Rules
Understanding Exponent Rules is crucial when multiplying terms with exponents. In algebra, when you multiply variables with exponents, you add the exponents together if they share the same base. For our example, consider:
  • \(2x^4 \times 3x^2\) gives us the term \(6x^{4+2} = 6x^6\)
When multiplying \(x^m\) and \(x^n\), the exponents \(m\) and \(n\) are added, resulting in \(x^{m+n}\). So, whenever a multiplication involves exponents, remember to use this rule to combine them properly. This simplifies expressions and makes solving them more systematic.
Combining Like Terms
Combining Like Terms is the step where things start to get simplified even further. After applying the distribution property and calculating each part, you might find terms that you can combine. In the example \(6x^6 - 14x^4\), each of the terms involved different powers of \(x\), and therefore:
  • The terms \(6x^6\) and \(-14x^4\) cannot be combined
This is because combining like terms requires the same variables raised to the same power. It's like stacking bricks: each brick has to be the same size to fit snugly. Understanding how to combine like terms is essential to simplify expressions and move closer to the final answer.

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Most popular questions from this chapter

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(3 \times 10^{7}\right)\left(8 \times 10^{4}\right)}{6 \times 10^{5}}$$

Expand and multiply. $$(x+7)^{2}$$

Simplify the following expressions. \(\left(4 y^{2}\right)^{3}\)

Multiply the following expressions. \(\left(6 a^{6} b^{3}\right)\left(3 a^{3} b^{6}\right)(a b)\)

Simplify the following expressions. \((2 a b)^{3}\)
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