To determine the missing exponent of [tex]\( y \)[/tex] in the second term of the polynomial [tex]\( 6xy^2 - 5x^2y^2 + 9x^2 \)[/tex] so that the polynomial becomes a trinomial with a degree of 3 after fully simplified, we need to analyze the highest degree term.
1. Identify the degrees of each term in the polynomial:
- The first term is [tex]\( 6xy^2 \)[/tex]:
- Degree of [tex]\( x \)[/tex] is 1.
- Degree of [tex]\( y^2 \)[/tex] is 2.
- Total degree of the term is [tex]\( 1 + 2 = 3 \)[/tex].
- The second term is [tex]\( -5x^2y^k \)[/tex] (where [tex]\( k \)[/tex] is the missing exponent we need to find):
- Degree of [tex]\( x^2 \)[/tex] is 2.
- Degree of [tex]\( y^k \)[/tex] is [tex]\( k \)[/tex].
- Total degree of the term is [tex]\( 2 + k \)[/tex].
- The third term is [tex]\( 9x^2 \)[/tex]:
- Degree of [tex]\( x^2 \)[/tex] is 2.
- There is no [tex]\( y \)[/tex] term, so degree contribution from [tex]\( y \)[/tex] is 0.
- Total degree of the term is [tex]\( 2 + 0 = 2 \)[/tex].
2. Determine the degree for the polynomial:
- We need the polynomial to be a trinomial with a highest degree of 3.
3. Find the missing exponent [tex]\( k \)[/tex]:
- For the term [tex]\( -5x^2y^k \)[/tex] to contribute to the degree of 3:
- The sum of the degrees (exponents) of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this term must add up to 3.
- The degree of [tex]\( x \)[/tex] is 2, so we need the degree of [tex]\( y \)[/tex] such that [tex]\( 2 + k = 3 \)[/tex].
4. Solve for [tex]\( k \)[/tex]:
- [tex]\( 2 + k = 3 \)[/tex]
- Subtract 2 from both sides:
[tex]\[
k = 3 - 2
\][/tex]
- So, [tex]\( k = 1 \)[/tex].
Hence, the missing exponent of [tex]\( y \)[/tex] in the second term [tex]\( -5x^2y^k \)[/tex] that will make the polynomial [tex]\( 6xy^2 - 5x^2y^2 + 9x^2 \)[/tex] a trinomial with a degree of 3 after fully simplified is [tex]\( 1 \)[/tex].
