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Chapter 5: Problem 97

Factor. $$ 60 x^{8} y^{6}+35 x^{4} y^{3}+5 $$

Short Answer

Expert verified
5(12 x^{8} y^{6} + 7 x^{4} y^{3} + 1)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Find the greatest common factor of the coefficients 60, 35, and 5. The GCF of these numbers is 5.
02

Factor out the GCF

Factor out the GCF from each term in the given expression:\[ 60 x^{8} y^{6} + 35 x^{4} y^{3} + 5 = 5(12 x^{8} y^{6} + 7 x^{4} y^{3} + 1) \]
03

Check for Further Factorization

Check if the polynomial inside the parentheses, \( 12 x^{8} y^{6} + 7 x^{4} y^{3} + 1 \), can be factored further. In this case, it cannot be factored further, so we stop here.

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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 number that can divide each term in a polynomial without leaving a remainder. In our exercise, we have the polynomial:
  • 60x8y6
  • 35x4y3
  • 5
We look at the coefficients (the numerical parts) of these terms: 60, 35, and 5. To find the GCF:
  1. List the factors of each coefficient:
    • 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
    • 35: 1, 5, 7, 35
    • 5: 1, 5
  2. Determine the largest factor that appears in all lists.
In this case, the GCF is 5. This is what we factor out from each term in the polynomial.
GCF
We found the GCF of the polynomial's coefficients to be 5. Now we will factor 5 out of each term. For our polynomial, this gives:
\[ 60 x^{8} y^{6} + 35 x^{4} y^{3} + 5 \]
Factoring out the GCF (5):
\[ 5 ( 12x^{8} y^{6} + 7x^{4} y^{3} + 1 ) \]
Each term inside the parentheses is now simplified, and the common factor 5 is factored out.
polynomial factorization
Polynomial factorization involves breaking down a complex polynomial into simpler factors. After factoring out the GCF, we check if the polynomial can be further factored. For:
\[ 12x^{8} y^{6} + 7x^{4} y^{3} + 1 \]
We look for any common factors or patterns, like the difference of squares, perfect square trinomials or other factoring techniques.
  • In our case, there are no further common factors.
  • It does not fit any special polynomial forms that we can factor easily.
Therefore, the factorization remains as\[ 5( 12x^{8} y^{6} + 7x^{4} y^{3} + 1 ) \]
This is the simplest form.

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