Answer :
To determine which statement is true about the factorization of the polynomial [tex]\( 30x^2 + 40xy + 51y^2 \)[/tex], let's evaluate each statement one by one.
### Statement 1:
"The polynomial can be rewritten after factoring as [tex]\( 10(3x^2 + 4xy + 5y^2) \)[/tex]."
Let's test this factoring:
[tex]\[ 10(3x^2 + 4xy + 5y^2) = 10 \cdot 3x^2 + 10 \cdot 4xy + 10 \cdot 5y^2 = 30x^2 + 40xy + 50y^2 \][/tex]
However, the original polynomial is [tex]\( 30x^2 + 40xy + 51y^2 \)[/tex], not [tex]\( 30x^2 + 40xy + 50y^2 \)[/tex]. Therefore, this statement is not true.
### Statement 2:
"The polynomial can be rewritten as the product of a trinomial and [tex]\( xy \)[/tex]."
To test this, we would need to factor the polynomial such that one factor is [tex]\( xy \)[/tex]. This would imply we should be able to find a polynomial such that:
[tex]\[ 30x^2 + 40xy + 51y^2 = xy (\text{some polynomial, assuming terms } x \text{ and } y \text{ correctly} \][/tex]
However, examining the polynomial, it is clear that none of its terms contain a factor of [tex]\( xy \)[/tex] in a straightforward manner. Therefore, this statement is not true.
### Statement 3:
"The greatest common factor of the polynomial is [tex]\( 51x^2y^2 \)[/tex]."
To check this, let's examine the terms in the polynomial:
[tex]\[ 30x^2, \quad 40xy, \quad 51y^2 \][/tex]
- [tex]\( 30x^2 \)[/tex] does not have [tex]\( y \)[/tex]
- [tex]\( 40xy \)[/tex] does not have [tex]\( x^2 \)[/tex]
- [tex]\( 51y^2 \)[/tex] does not have [tex]\( x \)[/tex] or [tex]\( x^2 y^2 \)[/tex]
Hence, [tex]\( 51x^2 y^2 \)[/tex] cannot be a factor of all the terms. Therefore, this statement is not true.
### Statement 4:
"The greatest common factor of the terms is 1."
Let's check the coefficients of the terms:
[tex]\[ 30, \quad 40, \quad 51 \][/tex]
The greatest common factor (GCF) of 30, 40, and 51 must be determined. We perform the prime factorization:
[tex]\[ 30 = 2 \times 3 \times 5 \][/tex]
[tex]\[ 40 = 2^3 \times 5 \][/tex]
[tex]\[ 51 = 3 \times 17 \][/tex]
There are no common prime factors among 30, 40, and 51. Therefore, the GCF of the coefficients is 1. Also, looking at the variables:
- [tex]\( x^2 \)[/tex] from [tex]\( 30x^2 \)[/tex]
- [tex]\( xy \)[/tex] from [tex]\( 40xy \)[/tex]
- [tex]\( y^2 \)[/tex] from [tex]\( 51y^2 \)[/tex]
None of these has a variable factor in common. Therefore, the GCF for the terms of the polynomial is indeed 1.
Given these evaluations, the true statement is:
The greatest common factor of the terms is 1.
So, the correct answer is Statement 4.
### Statement 1:
"The polynomial can be rewritten after factoring as [tex]\( 10(3x^2 + 4xy + 5y^2) \)[/tex]."
Let's test this factoring:
[tex]\[ 10(3x^2 + 4xy + 5y^2) = 10 \cdot 3x^2 + 10 \cdot 4xy + 10 \cdot 5y^2 = 30x^2 + 40xy + 50y^2 \][/tex]
However, the original polynomial is [tex]\( 30x^2 + 40xy + 51y^2 \)[/tex], not [tex]\( 30x^2 + 40xy + 50y^2 \)[/tex]. Therefore, this statement is not true.
### Statement 2:
"The polynomial can be rewritten as the product of a trinomial and [tex]\( xy \)[/tex]."
To test this, we would need to factor the polynomial such that one factor is [tex]\( xy \)[/tex]. This would imply we should be able to find a polynomial such that:
[tex]\[ 30x^2 + 40xy + 51y^2 = xy (\text{some polynomial, assuming terms } x \text{ and } y \text{ correctly} \][/tex]
However, examining the polynomial, it is clear that none of its terms contain a factor of [tex]\( xy \)[/tex] in a straightforward manner. Therefore, this statement is not true.
### Statement 3:
"The greatest common factor of the polynomial is [tex]\( 51x^2y^2 \)[/tex]."
To check this, let's examine the terms in the polynomial:
[tex]\[ 30x^2, \quad 40xy, \quad 51y^2 \][/tex]
- [tex]\( 30x^2 \)[/tex] does not have [tex]\( y \)[/tex]
- [tex]\( 40xy \)[/tex] does not have [tex]\( x^2 \)[/tex]
- [tex]\( 51y^2 \)[/tex] does not have [tex]\( x \)[/tex] or [tex]\( x^2 y^2 \)[/tex]
Hence, [tex]\( 51x^2 y^2 \)[/tex] cannot be a factor of all the terms. Therefore, this statement is not true.
### Statement 4:
"The greatest common factor of the terms is 1."
Let's check the coefficients of the terms:
[tex]\[ 30, \quad 40, \quad 51 \][/tex]
The greatest common factor (GCF) of 30, 40, and 51 must be determined. We perform the prime factorization:
[tex]\[ 30 = 2 \times 3 \times 5 \][/tex]
[tex]\[ 40 = 2^3 \times 5 \][/tex]
[tex]\[ 51 = 3 \times 17 \][/tex]
There are no common prime factors among 30, 40, and 51. Therefore, the GCF of the coefficients is 1. Also, looking at the variables:
- [tex]\( x^2 \)[/tex] from [tex]\( 30x^2 \)[/tex]
- [tex]\( xy \)[/tex] from [tex]\( 40xy \)[/tex]
- [tex]\( y^2 \)[/tex] from [tex]\( 51y^2 \)[/tex]
None of these has a variable factor in common. Therefore, the GCF for the terms of the polynomial is indeed 1.
Given these evaluations, the true statement is:
The greatest common factor of the terms is 1.
So, the correct answer is Statement 4.
