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Chapter 6: Problem 53

Factor each trinomial completely. $$ 25 z^{4}+5 z^{3}+z^{2} $$

Short Answer

Expert verified
z^2(25z^2 + 5z + 1)

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

Look for the greatest common factor in the trinomial. In the expression 25z^4 + 5z^3 + z^2, the GCF is z^2.
02

- Factor out the GCF

Factor out z^2 from each term: 25z^4 + 5z^3 + z^2 = z^2(25z^2 + 5z + 1).
03

- Recognize the Trinomial

Recognize that the expression inside the parentheses (25z^2 + 5z + 1) is already in its simplest form, and cannot be factored further over the integers. Therefore, the trinomial has been factored completely.

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Key Concepts

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

greatest common factor
The Greatest Common Factor (GCF) is the largest factor that divides all terms in an expression. Identifying the GCF is a crucial step in factoring trinomials. For the trinomial provided, 25z^4 + 5z^3 + z^2, we analyze each term:
  • For 25z^4, the factors are 1, 5, 25, z, z^2, z^3, z^4
  • For 5z^3, the factors are 1, 5, z, z^2, z^3
  • For z^2, the factors are 1, z, z^2
We identify the common factors: z, z^2. The greatest of these is z^2. Hence, the GCF of 25z^4 + 5z^3 + z^2 is z^2.
Knowing the GCF allows us to factor expressions more efficiently and sets the stage for further simplification.
factoring polynomials
Factoring polynomials involves rewriting the polynomial as a product of simpler polynomials. In our example, after identifying the GCF (z^2), we factor it out from each term in the trinomial:
25z^4 + 5z^3 + z^2 becomes z^2(25z^2 + 5z + 1).
Polynomials can often be factored into products of binomials or other polynomials, but sometimes they are already in their simplest form. Here:
25z^2 + 5z + 1 cannot be factored further over the integers.
To factor a polynomial, always start by finding the GCF and then see if the remaining polynomial can be broken down further.
simplifying expressions
Simplifying expressions is about making them as simple as possible. Once we factored out the GCF, the expression inside the parentheses was:
z^2(25z^2 + 5z + 1).
This expression is already simplified because 25z^2 + 5z + 1 has no further factors among the integers.
Simplifying involves:
  • Removing common factors
  • Combining like terms
  • Reducing fractions
For polynomials, simplifying usually involves factoring first. Always verify if the resulting polynomial or expression can be simplified further.Begin with basic steps like identifying the GCF and then proceed to inspect for any additional factors.

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