Final answer:
The missing first terms that, when added to the expression – 5xy^3 + 9x^2y, result in a binomial of degree 4 are 0, 8y^4, and 4xy^3.
Step-by-step explanation:
The question involves finding the missing term in an algebraic expression that simplifies to a binomial of degree 4. We need to choose terms which, when combined with the other terms in the expression, will result in a polynomial of degree 4 when simplified.
The given expression, minus the missing term, is:
– 5xy^3 + 9x^2y.
The degree of a term is the sum of the exponents of the variables in that term. The term –5xy^3 has a degree of 1 (from x) + 3 (from y^3), which is 4. The term –9x^2y has a degree of 2 (from x^2) + 1 (from y), which is 3.
To have a binomial of degree 4 after simplification, we can check each option:
0 - Adding zero would not change the degree of the expression.
5xy^3 - Adding it would cancel out the existing – 5xy^3, leaving a term of a lower degree, –9x^2y, which is not degree 4.
9x^2y - Adding it would increase the coefficient of the 9x^2y term but it would not make the expression a binomial of degree 4.
8y^4 - This term has a degree of 4, and adding it to the expression results in a binomial of degree 4 since the other term of degree 4 in the expression, –5xy^3, is not canceled out or altered in its degree.
4xy^3 - This term, when added to –5xy^3, would alter the coefficient but not cancel the term, resulting in a term of degree 4.
Thus, the missing terms that fit the criteria are 0, 8y^4, and 4xy^3.