To determine which factoring method can be considered for the polynomial [tex]\( 3x^3 + 5x + 6x^2 + 10 \)[/tex], it is essential to explore each factoring method:
1. Perfect-Square Trinomial: This method is applicable if the polynomial is of the form [tex]\( a^2x^2 + 2abx + b^2 \)[/tex]. Our polynomial has four terms, not in the form of a perfect-square trinomial. Therefore, this method is not applicable.
2. Difference of Squares: This method applies to polynomials of the form [tex]\( a^2 - b^2 \)[/tex]. Our polynomial does not fit this form, as it is not a binomial and does not represent a difference between two squares. Thus, this method is also not applicable.
3. Factor by Grouping: This technique involves grouping terms in pairs, factoring out the greatest common factor (GCF) from each pair, and then looking for a common factor in the resulting expression. Since our polynomial has four terms, this method could be suitable. We will explore this approach in more detail below.
4. Sum of Cubes: This method is applicable to expressions of the form [tex]\( a^3 + b^3 \)[/tex]. Our polynomial is not in this form, so this method is not applicable.
Factor by Grouping appears to be the most appropriate method for our polynomial [tex]\( 3x^3 + 5x + 6x^2 + 10 \)[/tex].
### Steps for Factor by Grouping
1. Group terms in pairs:
[tex]\[
(3x^3 + 6x^2) + (5x + 10)
\][/tex]
2. Factor out the GCF in each pair:
[tex]\[
3x^2(x + 2) + 5(x + 2)
\][/tex]
3. Factor out the common binomial factor ([tex]\(x + 2\)[/tex]):
[tex]\[
(x + 2)(3x^2 + 5)
\][/tex]
Thus, upon factoring by grouping, we obtain the factored form of the polynomial:
[tex]\[
(3x^3 + 5x + 6x^2 + 10) = (x + 2)(3x^2 + 5)
\][/tex]
So, the factoring method that can be considered for the polynomial [tex]\( 3x^3 + 5x + 6x^2 + 10 \)[/tex] is factor by grouping.
