To find the polynomial that represents the sum of the given polynomials, we need to add their corresponding coefficients. Let's break it down step-by-step.
We have two polynomials:
1. [tex]\( 3z^2 + 7z + 3 \)[/tex]
2. [tex]\( 5x^2 + 12z \)[/tex]
Firstly, note that the variables [tex]\( z \)[/tex] and [tex]\( x \)[/tex] are inconsistently used in the polynomials above. For the sake of consistent summation, we will presume they both represent the same variable, [tex]\( z \)[/tex]. Here are the steps to add these polynomials:
1. Identify the coefficients of the polynomials.
- For [tex]\( 3z^2 + 7z + 3 \)[/tex]:
- Coefficient of [tex]\( z^2 \)[/tex]: 3
- Coefficient of [tex]\( z \)[/tex]: 7
- Constant term: 3
- For [tex]\( 5x^2 + 12z \)[/tex] (treating [tex]\( x \)[/tex] as [tex]\( z \)[/tex]):
- Coefficient of [tex]\( z^2 \)[/tex]: 5
- There is no coefficient for the linear term ([tex]\( z \)[/tex]), so it is 0.
- Coefficient of [tex]\( z \)[/tex]: 12
- Constant term: There is no constant term, so it is 0.
2. Sum the corresponding coefficients.
- Sum of the coefficients of [tex]\( z^2 \)[/tex]: 3 (from the first polynomial) + 5 (from the second polynomial) = 8
- Sum of the coefficients of [tex]\( z \)[/tex]: 7 (from the first polynomial) + 12 (from the second polynomial) = 19
- Sum of the constant terms: 3 (from the first polynomial) + 0 (from the second polynomial) = 3
Putting it all together, the summed polynomial is:
[tex]\[ 8z^2 + 19z + 3 \][/tex]
Therefore, the polynomial that represents the sum is:
[tex]\[ 8x^2 + 19x + 3 \][/tex]
So the correct answer is:
A. [tex]\( 8x^2 + 19x + 3 \)[/tex]
