Chapter 4.2, Problem 52E

To determine

To find the next two monomials in the pattern; and GCF of all monomials in given pattern. 

Answer to Problem 52E

  $2x$

Explanation of Solution

Given Information: $2x,6x^2,18x^3,\mathrm{..}$

Formula used: Calculatethe next two monomials using geometric progression rule then use the GCF factorization rule to find GCF of the monomials.

Calculation:

PartI In the pattern $2x,6x^2,18x^3,\mathrm{..}$ divide 2nd term with 1st term to obtain a common ratio as can be seen each element is progressing in proportion.

  $\frac{6x^2}{2x}=3x$

And

  $\frac{18x^3}{6x^2}=3x$

This implies that each term is progressing by 3x times.

So the next two terms will be:

  $18x^3\times 3x=54x^4$

And

  $54x^4\times 3x=162x^5$

The next two terms are: $54x^4$ and $162x^5$ .

PartII To find the GCF of all monomials, take out the GCF of first two terms as the 1st term wholly contains in all other terms.

The factors of 2 are: 1, 2

The factors of 6 are: 1, 2, 3, 6

Factors of $x=x$

Factors of $x^2=x\times x$

Then the greatest common factor is $2x$