What is true about the sum of the two polynomials 6s²t − 2st² and 4s²t − 3st²?
The sum is a binomial with a degree of 3. Adding (6s²t − 2st²) + (4s²t − 3st²) combines like terms to give 10s²t − 5st². It has two terms (a binomial), and each term has degree 3 (2 + 1).
The answer
Add the two polynomials by combining like terms. Like terms have the exact same variables raised to the exact same powers.
- The s²t terms: 6s²t + 4s²t = 10s²t
- The st² terms: −2st² + (−3st²) = −5st²
So the sum is:
(6s²t − 2st²) + (4s²t − 3st²) = 10s²t − 5st²
Now classify the result. Count the terms and find the degree:
- Number of terms: two (10s²t and −5st²), so it is a binomial.
- Degree: the degree of a term is the sum of its exponents. In 10s²t the exponents are 2 (on s) and 1 (on t), giving 2 + 1 = 3. In −5st² the exponents are 1 and 2, also 1 + 2 = 3. The degree of the whole polynomial is the highest term degree, which is 3.
Therefore the true statement is: the sum is a binomial with a degree of 3.
Why the other choices are wrong
Typical distractors for this question include:
- "The sum is a trinomial with a degree of 3." Wrong on the term count. A trinomial has three terms. After combining like terms only two terms remain, so it is a binomial, not a trinomial. Students who forget to combine the like terms may mistakenly count four separate terms and then over-simplify.
- "The sum is a binomial with a degree of 2." Wrong on the degree. Degree is not the highest single exponent you see (2); it is the sum of the exponents within a term. Both terms sum to 3, so the degree is 3, not 2.
- "The sum is a monomial with a degree of 3." Wrong on term count. 10s²t and −5st² are not like terms (s²t ≠ st²), so they cannot be merged into a single term. Two unlike terms means a binomial.
The bigger picture
The key skill here is recognizing like terms in multivariable polynomials. It is tempting to treat 6s²t and −2st² as combinable because they share the same letters, but the exponents differ: s²t means s·s·t, while st² means s·t·t. Different exponent patterns = different terms that stay separate.
A reliable process for any polynomial sum:
- Drop the parentheses (watch the signs on subtracted terms).
- Group terms with identical variable-and-exponent patterns.
- Add the coefficients of each group.
- Count the surviving terms to name it (monomial = 1, binomial = 2, trinomial = 3).
- For each term, add its exponents; the largest such sum is the degree.
Applying that to 10s²t − 5st² gives two terms, each of degree 3 — a degree-3 binomial. Because both original polynomials were already degree-3 binomials with the same term structure, the sum stays a degree-3 binomial rather than growing in degree; adding polynomials never increases the degree.
What is true about the sum of the two polynomials 6s²t − 2st² and 4s²t − 3st²?
Frequently asked
How do you add two polynomials?
Remove the parentheses, keeping track of signs, then group and combine like terms — terms with identical variables and exponents. Add the coefficients of each group and leave unlike terms separate.
What are like terms in polynomials?
Like terms have exactly the same variables raised to exactly the same powers, differing only in their coefficients. For example, 6s²t and 4s²t are like terms, but s²t and st² are not because the exponents differ.
How do you find the degree of a polynomial term?
Add the exponents of all variables in that term. In 10s²t the exponents are 2 and 1, so the term's degree is 3. The polynomial's degree is the highest term degree.
What is a binomial?
A binomial is a polynomial with exactly two unlike terms, such as 10s²t − 5st². A monomial has one term and a trinomial has three.
Is 10s²t − 5st² a binomial or trinomial?
It is a binomial because it has two terms. The terms 10s²t and 5st² are not like terms, so they cannot combine into a single term.