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Chapter 9: Problem 21

Expand and simplify each expression. $$ -3 a(2 a-3) $$

Short Answer

Expert verified
\(-6a^2 + 9a\)

Step by step solution

01

Distribute the Constant

Distribute \(-3a\) through the parentheses \((2a - 3)\) using the distributive property, which means to multiply \(-3a\) by each term inside the parentheses.
02

Multiply the First Term

Multiply \(-3a\) by the first term inside the parentheses, \(2a\). This gives \( -3a \times 2a = -6a^2\).
03

Multiply the Second Term

Multiply \(-3a\) by the second term inside the parentheses, \(-3\). This gives \(-3a \times -3 = 9a\).
04

Combine the Results

Combine the results from the previous steps to form the expanded and simplified expression:\(-6a^2 + 9a\).

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Key Concepts

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

multiplication in algebra
Multiplication in algebra is a core concept that allows us to expand and simplify expressions. It involves multiplying constants, variables, or both. In the exercise, we leveraged the distributive property, which lets us multiply one term by each term inside parentheses.

To do this correctly, remember these steps:
  • Multiply the coefficients (numbers in front of variables).
  • Multiply the variables and use the rules of exponents.
In our example, \(-3a(2a-3)\), we distributed \(-3a\) by multiplying \(-3a \times 2a\) to get \[ -6a^2 \]. Then, we multiplied \(-3a \times -3\) to get \[ 9a \]. Notice how we kept the operations consistent by treating \(-3a\) as a single entity and applying it to both \[ 2a \] and \[-3\].
simplifying expressions
Simplifying expressions is all about making an expression as compact and easy to understand as possible. In the example above, after expanding, our result was \(-6a^2 + 9a\).

To simplify effectively, follow these tips:
  • Combine like terms: terms that have identical variables raised to the same power.
  • Ensure constants and coefficients are fully multiplied.
Simplification helps make complex problems more manageable and easier to work with in further algebraic operations. Here, after distributing and multiplying, no further combining of terms was needed, so \(-6a^2 + 9a\) is our simplified result.
polynomials
Polynomials are expressions made up of variables, coefficients, and constants, connected by addition, subtraction, and multiplication operations. Each separate expression is called a term. In our example, \(-6a^2 + 9a\) is a polynomial.

Key aspects of polynomials include:
  • Degrees: The highest power of the variable in the polynomial (here, the degree is 2).
  • Terms and coefficients: Segments of the polynomial (here, \(-6a^2 \) and \[ 9a \]).
Recognizing and working with polynomials is crucial in algebra, as they form the foundation for many other principles and operations. With practice and understanding, manipulating polynomials becomes straightforward.

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Most popular questions from this chapter

Challenge In Investigation \(2,\) you always assumed that two teams have the same chances of winning a single game. For this exercise, assume that Team \(\mathrm{A}\) has a 60\(\%\) chance of defeating Team \(\mathrm{B}\) in every game they play against each other. a. Suppose there is a one-game tournament between the teams and the winner of the game wins the tournament. What is the probability that Team \(\mathrm{A}\) will win? That Team \(\mathrm{B}\) will win? b. Use a tree diagram to show all the possibilities for the tournament. For example, in the first game, there are two branches: A wins or \(\mathrm{B}\) wins. (Hint: If \(\mathrm{A}\) wins the first two games, is a third game played?) c. Suppose the teams played \(1,000\) tournaments. In how many tournaments would you expect Team \(\mathrm{A}\) to win the first game? In how many of those tournaments would you expect Team \(\mathrm{A}\) to also win the second game? d. For each combination in your tree diagram, use similar reasoning to find the number of tournaments out of \(1,000\) you would expect to go that way. For example, one combination should be ABB; in how many tournaments out of \(1,000\) would you expect the winner to be \(\mathrm{A},\) then \(\mathrm{B}\) , and then \(\mathrm{B}\) ? (Hint. Check your answers by adding them; they should total to \(1,000 . )\) e. Find the total number of tournaments out of \(1,000\) in which each team wins the tournament. What is the probability that Team \(\mathrm{A}\) wins a tournament? f. Which tournament, one-game or best-two-out-of-three, is better for Team B?

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Suppose you roll two 12 -sided dice with faces numbered 1 to \(12 .\) a. How many possible number pairs can you roll? b. What is the greatest sum possible from a roll of two 12 -sided dice? c. What sum is most likely? What is the probability of this sum?
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