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Chapter 5: Problem 92

Simplify. $$ \left(3 x^{2}-4 x+6\right)-\left(-2 x^{2}+4\right)+(-5 x-3) $$

Short Answer

Expert verified
The simplified expression is \(5x^2 - 9x - 1\).

Step by step solution

01

Remove Parentheses

Begin by removing the parentheses. Remember to distribute any negative signs through the terms inside the parentheses: \( \begin{align*} \text{Expression:} & \big(3x^2 - 4x + 6\big) - \big(-2x^2 + 4\big) + \big(-5x - 3\big)\ \text{Becomes:} & 3x^2 - 4x + 6 + 2x^2 - 4 - 5x - 3 ewline ewline \end{align*} \)
02

Combine Like Terms

Next, combine like terms by grouping similar terms together. Like terms are terms that have the same variable raised to the same power: \( \begin{align*} 3x^2 + 2x^2 &- 4x - 5x + 6 - 4 - 3 ewline ewline 5x^2 & - 9x - 1 \end{align*} \)

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Key Concepts

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

Combining Like Terms
When simplifying algebraic expressions, it's important to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, in the expression given, we have several terms with the variable 'x' and some constants.

To combine like terms:
  • Identify terms with the same variable and exponent. In our case, the terms with \(x^2\) are \(3x^2\) and \(+2x^2\).
  • Next, look at the \(x\) terms: \(-4x\) and \(-5x\).
  • Finally, gather constant numbers: \(6,\) \(-4,\) and \(-3\).
After identifying, sum these terms separately:

\(3x^2 + 2x^2 = 5x^2\)
\(-4x - 5x = -9x\)
\(6 - 4 - 3 = -1\)

Then, you combine these results to get the simplified expression \( 5x^2 - 9x - 1 \). Mixing these terms leads us to our final combined expression.
Polynomials
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can have constants, variables, and exponents.

In our exercise, the given expression is a polynomial, \(3x^2 - 4x + 6 - (-2x^2+4) + (-5x-3)\). This is because it involves a sum of terms with the variable \(x\) raised to different powers and whole number coefficients.

Characteristics of polynomials include:
  • They consist of terms which are a combination of constants and variables.
  • The exponents of the variables are whole numbers.
  • Polynomials are classified based on their degree (the highest exponent). In our example, the degree is 2 from the term \(3x^2\).
Identifying a polynomial helps in understanding the terms to combine and simplifying the expression step by step.

Knowing how to handle polynomials is fundamental in simplifying algebraic expressions and solving them efficiently.
Distributive Property
The distributive property allows you to multiply a single term and two or more terms inside a parenthesis. It鈥檚 typically used in algebra to simplify expressions and solve equations.

In our original problem: \(3x^2 - 4x + 6 - (-2x^2 + 4) + (-5x - 3)\), we employed the distributive property to remove the parentheses.

Here鈥檚 how it鈥檚 applied:
  • For the term \(- (-2x^2 + 4)\), distributing the negative sign through the term changes the expression to \(+2x^2 - 4\).
  • That results in: \(3x^2 - 4x + 6 + 2x^2 - 4 - 5x - 3\).
Doing this step is crucial as it sets up the expression for the combining like terms step.

Understanding and applying the distributive property correctly allows you to manipulate and simplify expressions accurately, especially when dealing with parentheses and negative signs.

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Most popular questions from this chapter

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=x+\frac{2}{x-1}\\\ &g(x)=3 x^{3} \end{aligned} $$

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=64 x^{3}-8, g(x)=4 x-2 $$

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{26} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{x^{5}}{-3 y^{3}}\right)^{4} $$

Review factoring expressions and solving equations. $$ 4 x+9=0 $$
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