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Chapter 4: Problem 72

Find each product. $$(3 t-2)^{2}$$

Short Answer

Expert verified
The product is \( 9t^{2} - 12t + 4 \)

Step by step solution

01

Identify the binomial and apply the square of a binomial formula

Recognize that \( (3t - 2) \) is a binomial. To square it, apply the formula \( (a-b)^{2} = a^{2} - 2ab + b^{2} \). Substituting \( a = 3t \), and \( b = 2 \) in this formula we get \( (3t)^{2} - 2(3t)(2) + (2)^{2} \).
02

Simplify the terms

Now we simplify each term individually. We get \( 9t^{2} - 12t + 4 \)

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Key Concepts

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

Binomial Expansion
Expanding a binomial involves multiplying the terms within it, making them more manageable for calculations. Binomials are expressions made up of two distinct terms, such as \((3t - 2)\). When you expand a binomial raised to a power, you break it down into simpler components using specific formulas.
One such formula is the square of a binomial:
  • \((a-b)^2 = a^2 - 2ab + b^2\)
To use this in our exercise, we set \(a = 3t\) and \(b = 2\). By applying the formula, you systematically find each part of the expansion, ensuring no term is overlooked. This makes short work of what might seem like more complex algebra, by turning it into simple arithmetic operations.
Polynomials
Polynomials are algebraic expressions that consist of variables raised to non-negative integer exponents. Each term in the polynomial is a product of a constant coefficient and the variable raised to an exponent. In the result \(9t^{2} - 12t + 4\), each term fits this description.
Breaking it down:
  • \(9t^2\) represents a term where the variable \(t\) is squared and multiplied by 9.
  • \(-12t\) shows \(t\) multiplied by -12, resulting in a linear term.
  • 4 is a constant term, as it has no variable associated with it.
Polynomials are foundational in algebra since they enable a wide variety of operations and transformations, including simplification and factoring, which are vital tools in solving algebraic equations.
Exponents
Exponents, also known as powers, are a way to express repeated multiplication of a number by itself. For example, \( (3t)^2 \) means that \(3t\) is multiplied by itself once: \((3t) \times (3t) = 9t^2\).
In the context of the given exercise:
  • The exponent '2' signifies the operation of squaring \(3t\).
  • Exponents are essential in algebra because they represent larger operations compactly and allow for efficient computation.
Understanding exponents is crucial because they form the basis for more complex mathematical concepts, such as polynomial functions and equations. This knowledge is also useful for scientific notations, which frequently use powers of ten to represent very large or very small numbers compactly.

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