For the polynomial [tex]\(6xy^2 - 5x^2y^n + 9x^2\)[/tex] to be a trinomial with a degree of 3 after it has been fully simplified, what is the missing exponent of [tex]\(y\)[/tex] in the second term?

A. 0
B. 1
C. 2
D. 3



Answer :

To determine the missing exponent of [tex]\( y \)[/tex] in the second term of the polynomial [tex]\( 6xy^2 - 5x^2y^m + 9x^2 \)[/tex] so that it becomes a trinomial with a degree of 3 after it has been fully simplified, let's proceed with a step-by-step analysis.

1. Identifying the degrees of each term:
- The first term is [tex]\( 6xy^2 \)[/tex]:
- Degree of [tex]\( x \)[/tex] is 1.
- Degree of [tex]\( y \)[/tex] is 2.
- Total degree of this term = [tex]\( 1 + 2 = 3 \)[/tex].

- The third term is [tex]\( 9x^2 \)[/tex]:
- Degree of [tex]\( x \)[/tex] is 2.
- Since there is no [tex]\( y \)[/tex], its degree is 0.
- Total degree of this term = [tex]\( 2 + 0 = 2 \)[/tex].

2. Ensuring the polynomial is a trinomial with a degree of 3:
- Since we need the polynomial to have a degree of 3, the second term must also have a total degree of 3 when simplified.
- The second term is [tex]\( -5x^2y^m \)[/tex]:
- Degree of [tex]\( x \)[/tex] is already 2.
- We need the combined degree to be 3, so we set the equation:
[tex]\[ 2 + m = 3 \][/tex]

3. Solving for the missing exponent [tex]\( m \)[/tex]:
- Isolate [tex]\( m \)[/tex]:
[tex]\[ m = 3 - 2 \][/tex]
[tex]\[ m = 1 \][/tex]

Thus, the missing exponent of [tex]\( y \)[/tex] in the second term [tex]\( -5x^2y^m \)[/tex] must be [tex]\( \boxed{1} \)[/tex].