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Chapter 5: Problem 46

\(\left(7 x^{2}-15 x\right)-(2 x-3)\)

Short Answer

Expert verified
7x^2 - 17x + 3

Step by step solution

01

- Distribute the Negative Sign

Rewrite the expression \((7x^2 - 15x) - (2x - 3)\) by distributing the negative sign through the parentheses. This changes the signs of the terms inside the parentheses:\(= 7x^2 - 15x - 2x + 3\)
02

- Combine Like Terms

Combine the like terms in the expression \(7x^2 - 15x - 2x + 3\). The like terms are \( -15x \text{ and } -2x\):\(= 7x^2 - (15x + 2x) + 3\) \(= 7x^2 - 17x + 3\)

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Key Concepts

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

Distributing Negative Signs
In elementary algebra, one of the essential steps in simplifying polynomial expressions is distributing negative signs. When you see a negative sign outside parentheses, it means you apply the negative sign to each term inside. This process changes the signs of all the terms within the parentheses. For instance, in the expression \((7x^2 - 15x) - (2x - 3)\), the negative sign outside the second set of parentheses changes \(+2x\) to \(-2x\) and \(-3\) to \(+3\). Thus, the expression becomes \(7x^2 - 15x - 2x + 3\). It's crucial to handle negative signs properly to avoid mistakes in calculations.
Combining Like Terms
Once negative signs are distributed, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the expression \(7x^2 - 15x - 2x + 3\), the like terms are \(-15x\) and \(-2x\) because they both contain \(x\) raised to the first power. Combining \(-15x\) and \(-2x\) results in \(-17x\), simplifying the expression to \(7x^2 - 17x + 3\). Always make sure to keep track of the signs while combining like terms to avoid errors.
Polynomial Simplification
Simplifying a polynomial involves a series of steps such as distributing negative signs and combining like terms to present the polynomial in its simplest form. Simplification makes it easier to understand and work with polynomials in other mathematical problems. For instance, starting with \(7x^2 - 15x - 2x + 3\) and combining the like terms gives us \(7x^2 - 17x + 3\). This final, simplified form of the polynomial is more manageable and clear.
Elementary Algebra
Elementary algebra forms the foundation for much of the math you'll encounter later on. It includes basic operations and concepts such as combining like terms, distributing negative signs, and simplifying algebraic expressions. These skills are applied when working with polynomials like \(7x^2 - 15x - 2x + 3\). Through practice, understanding these basic principles will facilitate more advanced topics in algebra and beyond.
  • Start by identifying and understanding the essential operations like addition and subtraction within expressions.
  • Learn the rules for distributing negative signs to avoid common mistakes.
  • Combine like terms to simplify expressions effectively.
With these fundamentals, you'll be well-prepared for more complex problems.

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Most popular questions from this chapter

\(\left(x^{2}+10 x+16\right) \div(x+8)\)

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\(\left(x^{2}-10 x+16\right) \div(x-8)\)

Concentrated sulfuric acid is \(98 \%\) sulfuric acid. The liquid mixture in lead batteries is \(33.5 \%\) sulfuric acid. Use a system of linear equations to find the amount of concentrated sulfuric acid and the amount of water needed to make \(5 \mathrm{~L}\) of the liquid mixture for a lead battery. Round to the nearest tenth.

\(\left(w^{2}+9\right) \div(w+2)\)
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