Problem 74 - Beginning and Intermediate Algebra

Question

Perform the indicated operations. $$\begin{array}{l} \left(x^{2}-10 x-6\right) \\ -\left[\left(-8 x^{2}+11 x-1\right)+\left(5 x^{2}-9 x-3\right)\right] \end{array}$$

Step-by-Step Solution

Answer

The simplified expression after performing the indicated operations is \(-2x^2 - 8x - 4\).

Step 1: Distribute the negative sign to the polynomials inside the square brackets

Before performing the subtraction, we need to distribute the negative sign to each term inside the square brackets. \(-\left[\left(-8x^2 + 11x - 1\right) + \left(5x^2 - 9x - 3\right)\right] = \left[-\left(-8x^2 + 11x - 1\right)\right] - \left[5x^2 - 9x - 3\right] \) Now let's distribute the negative sign: \(\left[8x^2 - 11x + 1\right] - \left[5x^2 - 9x - 3\right]\)

Step 2: Subtract the polynomials

Now, subtract the second polynomial from the first polynomial: \(\left(x^2 - 10x - 6\right) - \left[8x^2 - 11x + 1\right] + \left[5x^2 - 9x - 3\right]\)

Step 3: Combine like terms

Combine the like terms (terms with the same power of x) in the expression: \(x^2 - 8x^2 + 5x^2 - 10x + 11x - 9x - 6 - 1 + 3\) Group and simplify: \((-2x^2) + (-8x) + (-4)\)

Step 4: Write the final simplified expression

The expression has been simplified as follows: \(-2x^2 - 8x - 4\) The final simplified expression is: \(-2x^2 - 8x - 4\)

Key Concepts

Distributive Property

In mathematics, the distributive property is a fundamental principle used to simplify expressions and equations. It allows us to "distribute" a factor across terms inside a bracket. This is particularly useful when dealing with polynomials. For example, if you have

• a term such as
  • \(- (a + b)\),
    the negative sign outside the bracket can be distributed to each term inside the bracket, resulting in
  • \(-a - b\).
• Applying this concept to the given exercise,
  • the negative sign must be applied to every term inside the brackets.
  • This changes each of \(-8x^2, 11x,\) and \(-1\) into \(8x^2, -11x, \) and \(+1\), respectively.

Understanding the distributive property helps in managing operations involving polynomials and is indispensable for simplifying expressions and solving equations easily.

Combining Like Terms

Combining like terms is a crucial step in the simplification of polynomial expressions. Like terms are terms within an expression that have identical variables raised to the same power. For instance:

• In an expression like \(3x^2 + 5x - 2x^2\),
  • the like terms are \(3x^2\) and \(-2x^2\) because they both have the variable \(x^2\).
  • To combine them, simply add or subtract their coefficients, resulting in \( (3 - 2)x^2 = x^2\).
• Similarly, apply the same process to other like terms.

In the exercise, once the expression has been simplified, we group and combine terms like \(-8x^2, 5x^2\),
as well as \(-10x, 11x, -9x\), and constants \(-6, -1,\) and \(+3\). This reduces the expression to a more manageable form:
\(-2x^2 - 8x - 4\). By doing this, expressions are simplified effectively.

Subtraction of Polynomials

Subtracting polynomials involves changing the sign of the terms in the polynomial to be subtracted and then following up with the combination of like terms. The process is akin to adding negative counterparts of each term.

• Let's break it down:
  • If you're subtracting
  • \((5x^2 + 3x - 7)\) from another polynomial,
  • the operation becomes adding \((-5x^2 - 3x + 7)\).
• This means reversing signs of all terms in the polynomial being subtracted.

In our exercise, the polynomial \(8x^2 - 11x + 1\) had to be subtracted, so it was rewritten with opposite signs \(-8x^2, 11x,\) and \(-1\).
Polynomials are easier to simplify when subtraction is approached this way. It turns a daunting task into a systematic step-by-step process.

Simplification of Expressions

Simplification is the final goal when working with polynomial expressions. By simplifying, we aim to express the equation in its simplest form, devoid of like terms and unnecessary complexity. The steps involved generally include:

• First, applying the distributive property to eliminate parentheses.
• Second, adding or subtracting like terms.

Upon applying these strategies and simplifying thoroughly, you reduce the expression from complicated and lengthy
to succinct and clear. In the exercise, the final expression achieved was \(-2x^2 - 8x - 4\).
When each term is accounted for and combined,
simplification turns what might seem a mess of variables and numbers into a tidy and manageable form.