To determine which terms could be the first term of an expression that creates a polynomial with a degree of 5, we'll analyze each term by finding its total degree. The polynomial degree is the highest sum of the exponents of the variables in each term. Let’s evaluate each term one by one:
1. Term: [tex]\( -4 x^3 y^2 \)[/tex]
- The exponent of [tex]\( x \)[/tex] is 3.
- The exponent of [tex]\( y \)[/tex] is 2.
- The total degree is [tex]\( 3 + 2 = 5 \)[/tex].
- Conclusion: This term has a degree of 5.
2. Term: [tex]\( x^3 \)[/tex]
- The exponent of [tex]\( x \)[/tex] is 3.
- There is no [tex]\( y \)[/tex] term, so the exponent of [tex]\( y \)[/tex] is 0.
- The total degree is [tex]\( 3 + 0 = 3 \)[/tex].
- Conclusion: This term does not have a degree of 5.
3. Term: [tex]\( 8.4 x^4 y^2 \)[/tex]
- The exponent of [tex]\( x \)[/tex] is 4.
- The exponent of [tex]\( y \)[/tex] is 2.
- The total degree is [tex]\( 4 + 2 = 6 \)[/tex].
- Conclusion: This term does not have a degree of 5.
4. Term: [tex]\( 5 x^4 y \)[/tex]
- The exponent of [tex]\( x \)[/tex] is 4.
- The exponent of [tex]\( y \)[/tex] is 1.
- The total degree is [tex]\( 4 + 1 = 5 \)[/tex].
- Conclusion: This term has a degree of 5.
5. Term: [tex]\( -x y^3 \)[/tex]
- The exponent of [tex]\( x \)[/tex] is 1.
- The exponent of [tex]\( y \)[/tex] is 3.
- The total degree is [tex]\( 1 + 3 = 4 \)[/tex].
- Conclusion: This term does not have a degree of 5.
6. Term: [tex]\( \frac{-2 x^4}{y} \)[/tex]
- The exponent of [tex]\( x \)[/tex] is 4.
- The exponent of [tex]\( y \)[/tex] in the denominator is equivalent to [tex]\( y^{-1} \)[/tex].
- The total degree is [tex]\( 4 + (-1) = 3 \)[/tex].
- Conclusion: This term does not have a degree of 5.
After evaluating all the terms, the terms that could be used to create a polynomial with a degree of 5 are:
- [tex]\( -4 x^3 y^2 \)[/tex]
- [tex]\( 5 x^4 y \)[/tex]
These are the terms that can start an expression resulting in a polynomial of degree 5.
