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Chapter 4: Problem 55

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

Short Answer

Expert verified
t^{3} - 9t^{2} + 27t - 27

Step by step solution

01

Expand the Expression

First, recognize that (t-3)^{3} is a binomial raised to the third power. This can be expanded using the binomial theorem: (a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}.
02

Identify the Terms

Identify the terms where a = t and b = 3. Substituting these terms into the binomial expansion formula.
03

Apply the Formula

Substitute a = t and b = 3 into the formula to get: (t - 3)^{3} = t^{3} - 3t^{2}(3) + 3t(3^{2}) - 3^{3}.
04

Simplify Each Term

Simplify each term of the expansion: t^3 - 3t^{2}(3) + 3t(9) - 27 which results in t^{3} - 9t^{2} + 27t - 27.

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Key Concepts

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

Expanding Binomials
When we talk about expanding binomials, we're using a formula to break down a binomial expression that's raised to a power. Learning to expand binomials involves understanding the Binomial Theorem.
The Binomial Theorem helps us expand expressions of the form \( (a + b)^n \).
Specifically, the theorem states that:
\[ (a + b)^n = \sum_{{k=0}}^n \binom{n}{k} a^{n-k} b^k \]
This may seem complex, but let's break it down step-by-step.
For our example, \((t - 3)^3\), the binomial theorem would look like this:
\[ (t - 3)^3 = t^3 - 3t^2 (3) + 3t (3^2) - 3^3 \]
Now that you've grasped the basics of expanding binomials, try applying it to new problems. Practice is key!
Polynomial Expressions
Polynomials are expressions involving variables and coefficients. In our example, the expanded form \(t^3 - 9t^2 + 27t - 27\) is a polynomial.
Polynomials can have multiple terms, constants, and variables of different degrees.
Here are things to notice in polynomials:
  • **Terms**: Parts of the expression separated by '+' or '-' signs.
  • **Degrees**: The exponents of the variables. In \(t^3\), the degree is 3.
  • **Coefficients**: The numbers in front of the variables. In \(-9t^2\), -9 is the coefficient.
Understanding the structure of polynomials helps in further operations like simplifying or solving equations.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means making them easier to work with. This often involves combining like terms and reducing the expression.
For the expanded polynomial \( t^3 - 9t^2 + 27t - 27 \), we make sure all terms are simplified:
  • **Combine Like Terms**: Ensure no further combination is possible. Like terms are those with the same variable and power.
  • **Simplify Each Term**: Perform any needed operations within each term.
  • **Order by Degree**: Sequence the terms by highest degree to lowest.

Practicing these steps will help you become more comfortable with algebra and much quicker at solving problems.

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Most popular questions from this chapter

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