Problem 36 Factor. $$21 t^{3}-10 t^{2}+t$... [FREE SOLUTION]

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Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 36

Factor. $$21 t^{3}-10 t^{2}+t$$

Short Answer

Expert verified

The factored form of the given polynomial is $t(21t^{2} - 10t + 1)$

Step by step solution

Identify common factor

In the given expression ($21 t^{3}-10 t^{2}+t$), we observe that every term has a factor of t. Thus, 't' is the common factor

Factor out the common term

Next, factor out the common factor 't'. Doing so will simplify the equation as follows: $t(21t^{2} - 10t + 1)$

Inspect for further factorisation

Inspect the bracketed polynomial ($21t^{2} - 10t + 1$). In this case, this polynomial cannot be further factorised since it does not meet the criteria for any further factorisation- rules i.e., it's not a perfect square, there's no grouping factorisation possible, nor difference of squares. Therefore, the simplified factored form is $t(21t^{2} - 10t + 1)$.

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Key Concepts

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

Common Factor

When working with polynomials, identifying the common factor is often the first step in simplification. A common factor is a term that is present in each part of the polynomial being examined. In algebra, this means looking for a variable, a number, or a combination of both that appears in every term.
For the expression given, $21t^3 - 10t^2 + t$, notice that 't' is present in every term:

$21t^3$ has the factor 't'
$-10t^2$ also includes 't'
$+t$ obviously contains 't'

Identifying 't' as a common factor allows you to pull it out of each term and simplify the polynomial effectively.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler, multiplied expressions. The first step is to extract the common factor, which we identified as 't' in the expression $21t^3 - 10t^2 + t$.

Once the common factor 't' is factored out, we are left with the internal polynomial $21t^2 - 10t + 1$. However, determining if this can be factored further involves checking for special cases:

Perfect Square: Each term does not meet the condition for forming a perfect square trinomial.
Grouping: The terms cannot be rearranged for factoring by grouping.
Difference of Squares: It is not structured as a difference of two squares.

In this scenario, the factorization ends here as $t(21t^2 - 10t + 1)$ is the simplest form.

Algebraic Expression Simplification

Simplifying algebraic expressions makes them easier to work with and understand. By reducing expressions to their simplest form, you can solve problems more efficiently and with greater accuracy.

In the case of our polynomial, simplifying started with identifying and factoring out the common factor 't'. This action alone reduced the polynomial complexity significantly. Although we stopped there in our factorization, this simplification is crucial.

Simplification steps:

Recognize terms that can be combined or factored together.
Extract common factors to condense the expression.
Evaluate if further polynomial simplification techniques apply.

Remember, simplifying an algebraic expression does not change the overall value, it only repositions components for clearer interpretation and problem-solving.

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91影视

Factor. $$21 t^{3}-10 t^{2}+t$$

Short Answer

Step by step solution

Identify common factor

Factor out the common term

Inspect for further factorisation

Key Concepts

Common Factor

Polynomial Factorization

Algebraic Expression Simplification

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