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Chapter 1: Problem 20

Perform the indicated multiplications. $$-2\left(-3 s t^{3}\right)(3 s-4 t)$$

Short Answer

Expert verified
The simplified expression is \(18s^2t^3 - 24st^4\).

Step by step solution

01

Distribute the First Factor

Start with multiplying the constant \(-2\) by the terms inside the parentheses. The expression is \(-2\left(-3st^3\right) = 2\cdot 3\cdot st^3 = 6st^3\).
02

Expand Using the Remaining Factor

Multiply the resulting \(6st^3\) by each term in the middle parentheses \((3s - 4t)\). This results in the products \(6st^3\cdot 3s\) and \(-6st^3\cdot 4t\).
03

Multiply Each Term

Calculate both products: \(6st^3\cdot 3s = 18s^2t^3\) and \(-6st^3\cdot 4t = -24st^4\).
04

Combine the Results

Combine the two results from Step 3: \(18s^2t^3 - 24st^4\). This is the complete simplified expression.

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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 comes in handy when dealing with expressions involving brackets. It tells us how to distribute or spread and multiply a single term over the terms within parentheses. In simpler words, to solve an expression like
  • \( a(b + c) \), you can multiply \( a \) with both \( b \) and \( c \).
This can be written as \( ab + ac \).
In our exercise, we start by applying this property to the constant
  • \(-2\)
with the expression inside the first set of parentheses
  • \( -3st^3 \), resulting in \( 6st^3 \).
The distributive property helps in simplifying algebraic expressions by eliminating parentheses and allowing easier manipulation of each component separately.
Polynomial Multiplication
Polynomial Multiplication involves multiplying terms from two or more polynomials. Each term of one polynomial is multiplied by every term of the other polynomial.
  • This process involves carefully distributing each term.
  • Keep track of the coefficients, bases, and exponents correctly.
In our case, we further distribute the resulting term \(6st^3\) by each term in the polynomial
  • \((3s - 4t)\).
Breaking it down, we multiply:
  • \(6st^3\) by \(3s\) to get \(18s^2t^3\),
  • and \(6st^3\) by \(-4t\) to get \(-24st^4\).
Multiplying polynomials requires careful calculation, especially when dealing with terms that have exponents.
Paying attention to the basic rules of algebra is crucial for accuracy.
Simplifying Expressions
Simplifying an algebraic expression involves reducing it to its simplest form. This means combining like terms and ensuring all operations are correctly executed.
  • The goal is to make the expression as short and easy to work with as possible.
In this problem, after the multiplication, we combined the resulting terms into
  • \(18s^2t^3 - 24st^4\).
Note that the terms are not like terms (because of different powers of \(s\) or \(t\)), so no further simplification can be made by adding or subtracting them.
Simplification allows the expression to be more easily interpreted and used in further calculations or problem-solving situations.

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