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Chapter 5: Problem 69

Simplify. $$ (12 x 2-34 x+14)-(32 x-34)+(54 x-12) $$

Short Answer

Expert verified
The simplified expression is \(12x^2 - 12x + 36\).

Step by step solution

01

Expand the Expression

The expression is \((12x^2 - 34x + 14) - (32x - 34) + (54x - 12)\). Distribute the negative sign in front of the second set of parentheses: \((12x^2 - 34x + 14) - 32x + 34 + 54x - 12\).
02

Combine Like Terms

Identify and group the like terms: - The \(x^2\) terms: \(12x^2\) - The \(x\) terms: \(-34x - 32x + 54x\)- The constant terms: \(14 + 34 - 12\).
03

Simplify the Expression

Calculate the sum for each group of like terms:- The \(x^2\) term remains as \(12x^2\).- Combine the \(x\) terms: \(-34x - 32x + 54x = -12x\).- Combine the constants: \(14 + 34 - 12 = 36\).Thus, the expression simplifies to \(12x^2 - 12x + 36\).

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Key Concepts

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

Combining Like Terms
One of the key steps in simplifying algebraic expressions is combining like terms. Like terms are terms that have the same variables raised to the same power. In our exercise, the expression contains terms like \(12x^2\), \(-34x\), \(-32x\), and \(54x\), along with constant terms such as 14, 34, and -12.

To combine like terms:
  • Identify terms with the same variables and exponents. For example, all the terms involving \(x\) should be grouped together.
  • Add or subtract the coefficients of these terms. For instance, the \(x\) terms \(-34x\), \(-32x\), and \(54x\) can be combined to \(-12x\) by performing the operations \(-34 - 32 + 54 = -12\).
  • For constant terms, simply add or subtract them as you would with regular numbers, such as \(14 + 34 - 12\), which simplifies to \(36\).
Combining like terms helps to simplify expressions into a more manageable form.
Distributing Negative Signs
Distributing negative signs is an important skill when working with algebraic expressions. Negative signs in front of parentheses must be distributed to each term within the parentheses. Consider the original expression \((12x^2 - 34x + 14) - (32x - 34) + (54x - 12)\).

To distribute the negative sign:
  • Focus on the set of parentheses that have a negative sign before them.
  • Multiply each term inside the parentheses by -1. In the example, \(-(32x - 34)\) becomes \(-32x + 34\).
  • Rewrite the expression with the newly adjusted terms: thus, you would have \((12x^2 - 34x + 14) - 32x + 34 + 54x - 12\).
The process of distributing negative signs ensures that all terms are correctly positioned for combining like terms.
Polynomials
Polynomials are algebraic expressions that consist of variables, coefficients, and constants combined using addition and subtraction. They come in various forms but generally include terms that are powers of variables like \(x\).

Polynomials can be identified by:
  • The number of terms they contain, such as monomials (one term), binomials (two terms), or trinomials (three terms).
  • The degree, which is the highest power of the variable in the polynomial. For instance, \(12x^2 - 12x + 36\) is a second-degree polynomial, with the term \(12x^2\) giving it this degree.
  • Their structure, which includes both like and unlike terms, as well as any present constants.
Understanding polynomials is essential for simplifying and solving various algebraic expressions, as they frequently appear in mathematical problems.

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