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Chapter 5: Problem 57

Factor each trinomial. \(12 p^{3}-12 p^{2}+3 p\)

Short Answer

Expert verified
The factored form is \(3p(4p^{2} - 4p + 1)\).

Step by step solution

01

- Identify the common factor

Look at the trinomial \(12p^{3} - 12p^{2} + 3p\) to find the common factor. All terms have a common factor of \(3p\).
02

- Factor out the greatest common factor

Factor out \(3p\) from each term in the trinomial. This yields: \[12p^{3} - 12p^{2} + 3p = 3p(4p^{2} - 4p + 1).\]
03

- Check the remaining trinomial

Ensure that the trinomial inside the parenthesis \(4p^{2} - 4p + 1\) cannot be factored further. Since it鈥檚 a quadratic trinomial and there are no two numbers that multiply to \(4 \times 1\) and add to \(-4\), it cannot be factored further.

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Key Concepts

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

greatest common factor
When you have an algebraic expression, finding the greatest common factor (GCF) is a crucial step. This involves identifying the largest factor that can divide each term of the expression.
For example, in the trinomial \(12p^{3} - 12p^{2} + 3p\), you will notice that all terms share common factors. Here are the steps to find the GCF:
  • Look at the coefficients (numbers in front of the variables): 12, 12, and 3. The largest number that can divide all three is 3.
  • Identify the smallest power of the variable \(p\) that is present in all terms. In this case, it is \(p\).
Combining these, the GCF for this trinomial is \(3p\). By factoring out \(3p\), you simplify the expression, which helps in solving or further factoring the equation.
quadratic trinomial
A quadratic trinomial has the form \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants. After factoring out the GCF from the initial expression \(12p^{3}-12p^{2}+3p\), we are left with \(4p^{2} - 4p + 1\), which is a quadratic trinomial.
Steps to factor quadratic trinomials include:
  • First, ensure the trinomial is in standard form.
  • Look for two numbers that multiply to \(a \times c\) (here, 4 in \(4p^{2} - 4p + 1\)) and add up to \(b\) (here, -4).
However, in our example, \(4p^{2} - 4p + 1\), there are no such pairs. This indicates that the trinomial cannot be factored further using simple methods.
algebraic expressions
Understanding algebraic expressions is essential for mastering algebra. These expressions involve variables and constants combined using operations such as addition, subtraction, multiplication, and division.
Consider the trinomial \(12p^{3}-12p^{2}+3p\). It contains:
  • Three terms: \(12p^{3}\), \(12p^{2}\), and \(3p\).
  • A variable: \(p\), which can represent an unknown number.
  • Constants: the numbers 12 and 3 represent known values.
By factoring out the GCF, we simplify the expression into more manageable parts, making it easier to solve or manipulate.
Breaking down complex algebraic expressions into simpler components, such as identifying and removing common factors, helps in performing operations and finding solutions efficiently.

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