Answer :
Sure! Let's look at the problem step-by-step.
We start with the polynomial expression:
[tex]\[ 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
We need to determine which of the given options, when added to this polynomial, will result in a 22nd-degree polynomial.
### Analyzing the Options:
Option A:
[tex]\[ 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 + 22 \][/tex]
- Here, the term [tex]\( 22 \)[/tex] is a constant (or [tex]\( 22x^0 \)[/tex]), which does not change the degree of the polynomial. Therefore, the highest degree term remains [tex]\( 14x^{19} \)[/tex].
Option B:
[tex]\[ 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 22x + 3 \][/tex]
- Adding [tex]\( 22x \)[/tex] introduces a term with degree 1, which still does not affect the highest degree term [tex]\( 14x^{19} \)[/tex].
Option C:
[tex]\[ x^{22} + 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
- This option adds the term [tex]\( x^{22} \)[/tex], which introduces a new term with degree 22. This becomes the highest degree in the polynomial.
Option D:
[tex]\[ 14x^{22} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
- This option adds the term [tex]\( 14x^{22} \)[/tex], also introducing a new term with degree 22. This becomes the highest degree in the polynomial.
### Conclusion:
Both options C and D introduce a term with degree 22, which would change the polynomial into a 22nd-degree polynomial. However, option C fits perfectly without changing the coefficient logic given in the problem.
So, the correct answer is:
[tex]\[ x^{22} + 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
We start with the polynomial expression:
[tex]\[ 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
We need to determine which of the given options, when added to this polynomial, will result in a 22nd-degree polynomial.
### Analyzing the Options:
Option A:
[tex]\[ 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 + 22 \][/tex]
- Here, the term [tex]\( 22 \)[/tex] is a constant (or [tex]\( 22x^0 \)[/tex]), which does not change the degree of the polynomial. Therefore, the highest degree term remains [tex]\( 14x^{19} \)[/tex].
Option B:
[tex]\[ 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 22x + 3 \][/tex]
- Adding [tex]\( 22x \)[/tex] introduces a term with degree 1, which still does not affect the highest degree term [tex]\( 14x^{19} \)[/tex].
Option C:
[tex]\[ x^{22} + 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
- This option adds the term [tex]\( x^{22} \)[/tex], which introduces a new term with degree 22. This becomes the highest degree in the polynomial.
Option D:
[tex]\[ 14x^{22} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
- This option adds the term [tex]\( 14x^{22} \)[/tex], also introducing a new term with degree 22. This becomes the highest degree in the polynomial.
### Conclusion:
Both options C and D introduce a term with degree 22, which would change the polynomial into a 22nd-degree polynomial. However, option C fits perfectly without changing the coefficient logic given in the problem.
So, the correct answer is:
[tex]\[ x^{22} + 14x^{19} - 9x^{15} + 11x^4 + 5x^2 + 3 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
