Question

which of the following expressions is a fifth - degree binomial?
a. $6x^{5}y^{4}-y^{5}x^{2}+6$
b. $17x^{5}y + 5xy^{6}$
c. $2xy^{2}-5y^{5}$
d. $x^{2}-2xy + 3x + 5y - 2$

Explanation:

Step1: Recall binomial and degree definitions

A binomial has 2 terms. The degree of a term with variables is the sum of exponents of variables. The degree of a polynomial is the highest degree of its terms.

Step2: Analyze Option A

Expression: \(6x^{5}y^{4}-y^{5}x^{2}+6\)
Number of terms: 3 (trinomial), so eliminate A.

Step3: Analyze Option B

Expression: \(17x^{5}y + 5xy^{6}\)
Degree of first term: \(5 + 1=6\), second term: \(1 + 6 = 7\). Highest degree is 7, so not fifth - degree. Eliminate B.

Step4: Analyze Option C

Expression: \(2xy^{2}-5y^{5}\)
Number of terms: 2 (binomial).
Degree of first term: \(1+2 = 3\), second term: 5. Highest degree is 5. So this is a fifth - degree binomial.

Step5: Analyze Option D

Expression: \(x^{2}-2xy + 3x+5y - 2\)
Number of terms: 5 (polynomial with 5 terms), eliminate D.

Answer:

C. \(2xy^{2}-5y^{5}\)