To determine which statement is true about the polynomial [tex]\( -3x^4y^3 + 8xy^5 - 3 + 18x^3y^4 - 3xy^5 \)[/tex] after it has been fully simplified, we will follow these steps:
1. Combine Like Terms:
We start by identifying and combining like terms in the polynomial. In this case, the terms [tex]\( 8xy^5 \)[/tex] and [tex]\( -3xy^5 \)[/tex] are like terms because they contain the same variables raised to the same powers.
[tex]\[
8xy^5 - 3xy^5 = (8 - 3)xy^5 = 5xy^5
\][/tex]
So, the simplified polynomial is:
[tex]\[
-3x^4y^3 + 5xy^5 - 3 + 18x^3y^4
\][/tex]
2. Count the Number of Terms:
The simplified polynomial now has four terms:
[tex]\[
-3x^4y^3, \quad 5xy^5, \quad -3, \quad 18x^3y^4
\][/tex]
3. Determine the Degrees of Each Term:
The degree of a term is the sum of the exponents of the variables in the term.
- For [tex]\( -3x^4y^3 \)[/tex]: The degree is [tex]\( 4 + 3 = 7 \)[/tex].
- For [tex]\( 5xy^5 \)[/tex]: The degree is [tex]\( 1 + 5 = 6 \)[/tex].
- For [tex]\( -3 \)[/tex]: The degree is [tex]\( 0 \)[/tex] (since it is a constant term).
- For [tex]\( 18x^3y^4 \)[/tex]: The degree is [tex]\( 3 + 4 = 7 \)[/tex].
4. Find the Maximum Degree:
The degrees of the terms are [tex]\( 7, 6, 0, \)[/tex] and [tex]\( 7 \)[/tex]. The maximum degree among these is [tex]\( 7 \)[/tex].
5. Conclusion:
- The polynomial [tex]\( -3x^4y^3 + 5xy^5 - 3 + 18x^3y^4 \)[/tex] has 4 terms.
- The highest degree of the polynomial is 7.
Therefore, the correct statement is:
- It has 4 terms and a degree of 7.
