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Chapter 6: Problem 185

Multiply. $$-x(x-10)$$

Short Answer

Expert verified
-x^2 + 10x

Step by step solution

01

- Distribute the Negative Sign

First, recognize that the negative sign in front of the x will need to be distributed to each term inside the parentheses. Write the expression as \(-x \cdot (x - 10)\)
02

- Multiply \(-x\) with \(x\)

Next, multiply \(-x\) by the first term inside the parentheses which is \(x\). This results in\(-x \cdot x = -x^2\)
03

- Multiply \(-x\) with \(-10\)

Then, multiply \(-x\) by the second term inside the parentheses which is \(-10\). Remember that multiplying two negative numbers results in a positive value. This results in \(-x \cdot (-10) = 10x\)
04

- Combine the Results

Combine the results from Steps 2 and 3. The final expression is \(-x^2 + 10x\)

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Key Concepts

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

Multiplying Variables
When you multiply variables, you perform an operation similar to multiplying numbers. If the variables are the same, like x and x, you combine them by adding their exponents. For example, multiplying \(x \cdot x\), results in \(x^2\). This is because \(x^1 \cdot x^1 = x^{1+1} = x^2\).

This becomes important when dealing with polynomials or expressions that include variables. Always remember to combine like terms by adding the exponents, but only for the variables with the same base.
Negative Numbers
Working with negative numbers follows its own set of rules that can sometimes be tricky. One of the key rules to remember is that multiplying two negative numbers results in a positive number, such as \(-1 \cdot -1 = 1\).

Conversely, multiplying a positive number by a negative number will result in a negative number. For example, \(4 \cdot -3 = -12\).

In the context of expressions, negative signs are important. They can change the entire value of the term they are attached to. So always pay special attention to the signs when performing arithmetic with negative numbers.
Polynomial Expressions
Polynomial expressions are sums of terms where each term includes a variable raised to an exponent and multiplied by a coefficient. For example, \(-x^2 + 10x\) is a polynomial expression with two terms: \(-x^2\) and \(10x\).

When multiplying polynomials, you often need to use the distributive property to ensure each term inside the parentheses is multiplied by the term outside. For example, multiplying \-x(x - 10)\ involves distributing the \-x\ to both terms \x\ and \-10\.

Always combine the like terms if needed, and simplify the expression to its final form. Be diligent with the signs, as they can impact the result significantly.

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Most popular questions from this chapter

Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression \(\frac{n^{2}-n}{2},\) where \(n\) represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

Divide the monomials. $$\frac{\left(6 a^{4} b^{3}\right)\left(4 a b^{5}\right)}{\left(12 a^{2} b\right)\left(a^{3} b\right)}$$

Simplify. (a) \(4^{-2}\) (b) \(10^{-3}\)

Mixed Practice (a) \(p^{4} \cdot p^{6}\) (b) \(\left(p^{4}\right)^{6}\)

Simplify. (a) \((-5)^{-2}\) (b) \(-5^{-2}\) (c) \(\left(-\frac{1}{5}\right)^{-2}\) (d) \(-\left(\frac{1}{5}\right)^{-2}\)
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