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Chapter 1: Problem 49

Perform the indicated operations and simplify. $$(2 x-5)\left(x^{2}-x+1\right)$$

Short Answer

Expert verified
The simplified result is \(2x^3 - 7x^2 + 7x - 5\).

Step by step solution

01

Distribute the first term

Start by distributing the first term, \(2x\), across the expression inside the parentheses. Multiply \(2x\) by each term in \(x^2 - x + 1\). This results in \(2x \cdot x^2 - 2x \cdot x + 2x \cdot 1\), which simplifies to \(2x^3 - 2x^2 + 2x\).
02

Distribute the second term

Next, distribute the second term, \(-5\), across the expression inside the parentheses. Multiply \(-5\) by each term in \(x^2 - x + 1\). This results in \(-5 \cdot x^2 + 5 \cdot x - 5 \cdot 1\), which simplifies to \(-5x^2 + 5x - 5\).
03

Combine like terms

Now, combine the results from the previous steps: \(2x^3 - 2x^2 + 2x - 5x^2 + 5x - 5\). Group the like terms together: \(2x^3 + (-2x^2 - 5x^2) + (2x + 5x) - 5\), which simplifies to \(2x^3 - 7x^2 + 7x - 5\).
04

Write the simplified expression

The expression is now fully simplified. After combining all like terms, we have the final simplified expression: \(2x^3 - 7x^2 + 7x - 5\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that allows you to multiply a single term by each term inside a parenthesis. It鈥檚 like spreading a multiplying factor across each part of the expression. In this problem, we have two separate applications of the distributive property:

  • First Application: The initial task is to distribute the term \(2x\) to each term inside the parentheses \((x^2 - x + 1)\). Think of it as a mini multiplication for each term:
    • Multiply \(2x\) by \(x^2\) to get \(2x^3\).
    • Multiply \(2x\) by \(-x\) to get \(-2x^2\).
    • Multiply \(2x\) by \(1\) to get \(2x\).
  • Second Application: After distributing \(-5\) in the same manner:
    • Multiply \(-5\) by \(x^2\) to get \(-5x^2\).
    • Multiply \(-5\) by \(-x\) to get \(5x\).
    • Multiply \(-5\) by \(1\) to get \(-5\).
By using the distributive property, we have broken down the multiplication process into simpler steps, making it easier to handle. This method is really useful when working with polynomials and sets us up perfectly for the next stage, which is combining like terms.
Combining Like Terms
Combining like terms is an essential skill that helps simplify expressions by uniting terms with the same variable part. Think of it as collecting similar items in a group. After distributing, we have the expression: \(2x^3 - 2x^2 + 2x - 5x^2 + 5x - 5\).

Here鈥檚 how we combine like terms in this equation:
  • Identify Like Terms: Look for terms with identical variable parts. In this case:
    • \(x^3\) term: \(2x^3\)
    • \(x^2\) terms: \(-2x^2\) and \(-5x^2\)
    • \(x\) terms: \(2x\) and \(5x\)
    • Constant terms: \(-5\)
  • Combine the Terms:
    • Combine the \(x^2\) terms: \(-2x^2 + (-5x^2) = -7x^2\)
    • Combine the \(x\) terms: \(2x + 5x = 7x\)
Finally, we write the expression with combined like terms: \(2x^3 - 7x^2 + 7x - 5\). Combining like terms is like tidying up your work, ensuring the expression is as simple as it can be. It makes it easier to read and further manipulate in equations.
Simplifying Expressions
Simplifying expressions is about making complex equations less cluttered and more straightforward. The end goal is to write an expression in its simplest form, just like solving a puzzle until you have the simplest picture.

When simplifying an expression, here's the approach:
  • Step 1: Use the distributive property to expand any multiplication over addition or subtraction, as we did earlier.
  • Step 2: Identify and combine like terms. This involves looking for similar variables and powers, uniting them to simplify the equation.
  • Step 3: Arrange the expression neatly, often with terms organized from highest to lowest powers for clarity.
By following these steps with our example, we obtained the final simplified expression: \(2x^3 - 7x^2 + 7x - 5\). Simplification takes potentially bulky expressions and makes them manageable, ensuring we can easily interpret and work with them in further math problems.

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Most popular questions from this chapter

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt{x^{5}}\) (b) \(\sqrt[4]{x^{6}}\)

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt[4]{b^{3}} \sqrt{b}\) (b) \((2 \sqrt{a})(\sqrt[3]{a^{2}})\)

Write the number indicated in each statement in scientific notation. (a) A light-year, the distance that light travels in one year, is about \(5,900,000,000,000 \mathrm{mi}\) (b) The diameter of an electron is about \(0.0000000000004 \mathrm{cm}\) (c) A drop of water contains more than 33 billion billion molecules.

Decimal Notation Write each number in decimal notation. (a) \(3.19 \times 10^{5}\) (b) \(2.721 \times 10^{8}\) (c) \(2.670 \times 10^{-8}\) (d) \(9.999 \times 10^{-9}\)

Test the equation for symmetry. $$y=x^{2}+|x|$$
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