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Chapter 6: Problem 46

Multiply. $$-0.14\left(0.9 n^{2}-40 n-8.7\right)$$

Short Answer

Expert verified
Question: Simplify the expression -0.14(0.9 n虏 - 40 n - 8.7). Answer: -0.126 n虏 + 5.6n + 1.218

Step by step solution

01

Multiply the first term of the polynomial by -0.14

To do this, we will multiply -0.14 by the first term of the polynomial, which is 0.9 n虏: $$-0.14 \times 0.9 n^{2}$$ After multiplying, we get: $$-0.126 n^{2}$$
02

Multiply the second term of the polynomial by -0.14

Next, we will multiply -0.14 by the second term of the polynomial, which is -40 n: $$-0.14 \times (-40 n)$$ After multiplying, we get: $$5.6n$$
03

Multiply the third term of the polynomial by -0.14

Finally, we will multiply -0.14 by the third term of the polynomial, which is -8.7. $$-0.14 \times (-8.7)$$ After multiplying, we get: $$1.218$$
04

Combine the terms

Now that we have the simplified terms after the multiplication, we combine them as follows: $$-0.126 n^{2} + 5.6n + 1.218$$
05

Final Answer

Simplifying the given expression, we obtain: $$-0.126 n^{2} + 5.6n + 1.218$$

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that helps us simplify expressions. It allows us to multiply a single term by each term inside a bracket, distributing the multiplication across the terms. This is crucial for polynomial multiplication. In mathematical terms, the distributive property is expressed as:
  • \( a(b + c) = ab + ac \)
This means that a term outside the parentheses is multiplied by each term inside separately. In our example, we apply this by multiplying \(-0.14\) with each term inside the polynomial \((0.9n^2 - 40n - 8.7)\). This step-by-step distribution helps ensure we don鈥檛 miss any terms and makes simplification straightforward.
Mathematical Operations
Mathematical operations in algebra involve various processes such as addition, subtraction, multiplication, and division. When performing polynomial multiplication, specifically using the distributive property, we focus primarily on multiplication.
  • Each step involves multiplying each term in the polynomial by the constant or variable outside the bracket.
For the exercise, we carried out multiplication three times:
  • \(-0.14 \times 0.9n^2\)
  • \(-0.14 \times (-40n)\)
  • \(-0.14 \times (-8.7)\)
Remember that when multiplying negative numbers, two negatives make a positive. This is why we saw a positive result in some multiplications. Finally, we sum the individual products to arrive at the proper polynomial.
Algebraic Expressions
Algebraic expressions are a way of expressing mathematical ideas using variables and constants. They can include operations such as addition, subtraction, and multiplication. In this exercise, the polynomial \(0.9n^2 - 40n - 8.7\) is an algebraic expression composed of:
  • Multiple terms: \(0.9n^2\), \(-40n\), and \(-8.7\)
  • Different coefficients and powers of the variable \(n\)
Every term involves a coefficient (the number in front of the variable) and at times, an exponent (showing powers of the variable). Distributing and combining these terms through mathematical operations simplifies the expression to a new form. It's crucial in understanding equations and functions where using expressions to represent real-world scenarios is necessary.

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Most popular questions from this chapter

Complete the table. Do not solve. A grocery store sells two different boxes of a particular cereal. The large box sells for 4.59 dollars the small box, for 3.79 dollars. In one day, the store sold 32 more small boxes than large boxes. $$\begin{array}{l|l|l|l}\text { Categories } & \text { Selling price } & \text { Number of boxes sold } & \text { Income } \\\\\hline \text { Small box } & & & \\\\\hline \text { Large box } & & & \\\\\hline\end{array}$$

Use \(<\) or \(>\) to make a true statement. $$.2 .891 ? 2.8909$$

Evaluate each expression using the given values. $$v t+\frac{1}{2} a t^{2} ; v=35, t=1.2, a=10$$

Solve. The current in a circuit is measured to be \(-3.5\) amperes. If the resistance is 0.8 ohm, what is the voltage? (Use \(V=i r\).)

Evaluate each expression using the given values. $$v t+\frac{1}{2} a t^{2} ; v=20, t=0.6, a=-12.5$$
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