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Chapter 6: Problem 198

Multiply. $$4 x^{3}\left(x^{4}-3 x+7\right)$$

Short Answer

Expert verified
The product is \(4x^7 - 12x^4 + 28x^3\).

Step by step solution

01

- Distribute the term

Distribute the term outside the parentheses, which is \(4x^3\), to each term inside the parentheses \(x^4 - 3x + 7\).
02

- Multiply each term

Multiply \(4x^3\) by each term inside the parentheses separately. - \(4x^3 \times x^4\) - \(4x^3 \times -3x\) - \(4x^3 \times 7\)
03

- Simplify the multiplication

Simplify each term: - \(4x^3 \times x^4 = 4x^{3+4} = 4x^7\) - \(4x^3 \times -3x = -12x^{3+1} = -12x^4\) - \(4x^3 \times 7 = 28x^3\)
04

- Combine the results

Combine the simplified terms to form the final expression: \(4x^7 - 12x^4 + 28x^3\)

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Key Concepts

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

Distribution
In polynomial multiplication, the distribution property is a vital tool. Distribution allows us to multiply a single term outside parentheses with each term inside the parentheses.
Let's take the expression: \[ 4x^{3}\big(x^{4}-3x+7\big) \]Distribute the outer term, \[ 4x^{3} \]to every term inside the parentheses:
  • \[ 4x^3 \times x^4 \]
  • \[ 4x^3 \times (-3x) \]
  • \[ 4x^3 \times 7 \]
By following this rule carefully, you ensure that none of the terms are left out. This step sets the stage for accurate multiplication and further simplification.
Exponent Rules
Understanding exponent rules is critical when multiplying polynomials. The core rule needed here is: when multiplying powers of the same base, you add the exponents.For example:\[ a^{m} \times a^{n} = a^{m+n} \]Let's apply this rule to the three terms we distributed previously:
  • \[ 4x^3 \times x^4 = 4x^{3+4} = 4x^7 \]
  • \[ 4x^3 \times (-3x) = -12x^{3+1} = -12x^4 \]
  • \[ 4x^3 \times 7 = 28x^3 \](Multiplying by a constant doesn't change the exponent)
These calculations are straightforward when you remember to add the exponents for terms with the same base.
Simplifying Expressions
After distributing and applying exponent rules, the final step is to simplify the expression. This involves combining all the simplified terms into one polynomial:\[ 4x^7 - 12x^4 + 28x^3 \]When simplifying, ensure each term is ordered by descending powers of the variable if possible. This makes the final expression neat and easy to interpret. Always double-check your work to ensure no terms have been missed or incorrectly simplified. This final step pulls all your hard work together into a presentable format.

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