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Chapter 1: Problem 35

Perform the indicated multiplications. $$-3(3-2 T)(3 T+2)$$

Short Answer

Expert verified
The result of the multiplication \\(-3(3-2T)(3T+2)\\) is \\(-9T^2 - 9\\).

Step by step solution

01

Apply Distribution Inside Parentheses

First, we apply the distributive property by multiplying each term inside the first parentheses \(3 - 2T\) by each term inside the second parentheses \(3T + 2\). We start with \(-3T(3T + 2)\): \[-3T(3T + 2) = -3T \cdot 3T + (-3T) \cdot 2 = -9T^2 - 6T\]
02

Continue Distribution

Next, distribute \(-3\) across the terms in \(3T + 2\): \[-3(3 - 2T)(3T + 2) = -3(3T) + (-3)(2) = -9 + 6T\]
03

Combine Like Terms

Combine the terms from Step 1 and Step 2: The terms are \-9T^2\, \-6T\, \6T\, and \-9\. Combine \-6T\ with \6T\ to simplify: \[-9T^2 - 6T + 6T - 9 = -9T^2 - 9\]
04

Conclusion

After simplifying the expression by combining like terms, we obtain the final result: The expression \(-3(3 - 2T)(3T + 2)\) simplifies to \(-9T^2 - 9\).

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication is a process used to simplify expressions that involve two or more polynomials being multiplied together. In our exercise, we have a product of the terms: \[ -3(3 - 2T)(3T + 2) \]. The objective is to multiply these values in a systematic way.
  • To start, note how each term in one set of parentheses must be multiplied by each term in the other parentheses.
  • This means you need to carry out operations on pairs of terms, strategically working through the polynomial multiplication.
By doing this, we essentially expand the expression, readying it for combining and simplifying. Remember, the order doesn't matter due to the commutative property of multiplication: you can multiply in whatever sequence you find most straightforward, but consistency is key for accuracy.
Distributive Property
The distributive property is a useful algebraic property that helps to simplify expressions and multiply terms across a sum or difference inside parentheses. It can be expressed mathematically as \( a(b+c) = ab + ac \).
  • In our exercise, the distributive property is first applied to the expression \(3 - 2T\) times \(3T + 2\) which results in two parts that need further expansion."
  • Secondly, the distributive property helps in addressing the multiplication by \(-3\). Each term resulting from the previous multiplication is multiplied by this factor.
The application of this property is essential for breaking down complex expressions to manageable terms, allowing us to eventually simplify the entire expression.
Combining Like Terms
Combining like terms is a critical step when simplifying algebraic expressions. It involves summing up terms that have identical variable components. Consider terms as 'like' if they contain the same variable raised to the same power.
  • In our specific case, we encounter terms such as \(-9T^2\), \(-6T\), and \(+6T\).
  • Here, \(-6T\) and \(+6T\) are like terms because they both include the variable \(T\) to the first power. They effectively cancel each other because their sum equals zero.
After simplifying by combining, you are left with the final expression \(-9T^2 - 9\). This process is crucial as it significantly eases the complexity of algebraic expressions, permitting us to arrive at simpler and cleaner solutions.

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Most popular questions from this chapter

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