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Chapter 6: Problem 44

Factor each trinomial. $$ 45 t^{3}+60 t^{2}+20 t $$

Short Answer

Expert verified
5t(9t^2 + 12t + 4)

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

Examine the coefficients and the variable parts of each term. The coefficients are 45, 60, and 20. The largest number that divides all these coefficients is 5. The smallest power of the variable present in all terms is t. Therefore, the GCF is 5t.
02

- Factor Out the GCF

Factor out the GCF (5t) from each term in the trinomial. This gives: \[ 45t^3 + 60t^2 + 20t = 5t(9t^2 + 12t + 4) \]
03

- Simplify the Remaining Trinomial (if possible)

Check if the remaining trinomial inside the parentheses can be factored further. In this case: \[ 9t^2 + 12t + 4 \] The trinomial does not factor further using simple methods, hence this is the final factorization.

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Key Concepts

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

Greatest Common Factor
To start factoring any algebraic expression, especially trinomials, it is essential to find the Greatest Common Factor (GCF). The GCF is the largest number and variable that divides all terms in the expression.

For example, in the trinomial given, 45t鲁 + 60t虏 + 20t, you need to look at both the coefficients (45, 60, and 20) and the variables (t鲁, t虏, and t).
  • First, find the GCF of the coefficients, which is 5 because it is the largest number that divides 45, 60, and 20.
  • Next, determine the smallest power of the variable t present in all terms, which is t.
    This gives us the GCF of 5t.
Factoring out the GCF is a crucial step and simplifies the process of trinomial factoring.
Trinomial Factoring
Once you have identified and factored out the GCF, the next step is to focus on the remaining trinomial. In the provided exercise, after factoring out 5t, the trinomial becomes:
5t(9t虏 + 12t + 4).

Now you need to check if the resulting trinomial inside the parentheses can be factored further.
  • For some trinomials, using methods like the quadratic formula or AC method can help factor them further.
  • However, in this case, 9t虏 + 12t + 4 does not factor further using simple methods.
Remember, sometimes the remaining trinomial may already be in its simplest form, and no further factoring is possible.
Algebraic Expressions
Understanding algebraic expressions is fundamental to mastering topics like factoring trinomials. Algebraic expressions consist of numbers, variables, and operations.

Expressions like 45t鲁 + 60t虏 + 20t contain:
  • Coefficients (numerical parts like 45, 60, 20)
  • Variables (like t with different exponents)
When factoring an expression, your goal is to simplify or rewrite it in an equivalent but easier-to-handle form.

By factoring out the GCF or further breaking down the trinomial, you can solve equations, simplify expressions, and better understand the underlying algebraic structure.
This step-effective approach is key to solving more complex algebraic problems efficiently.

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