Question Video: Identifying Trinomials from a List of Expressions | Nagwa Question Video: Identifying Trinomials from a List of Expressions | Nagwa

# Question Video: Identifying Trinomials from a List of Expressions Mathematics • First Year of Preparatory School

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Consider this list of expressions: A. −1 − 𝑥 + 1. B. 𝑥 + 𝑦. C. 2𝑥𝑦 − 3𝑥⁴ + 1. D. 𝑥² + 𝑥^(1/3) + 2𝑥. E. −3𝑥 + 2𝑥 + 0. Which expression (or expressions) is a trinomial?

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### Video Transcript

Consider this list of expressions. (A) Negative one minus 𝑥 plus one. (B) 𝑥 plus 𝑦. (C) Two 𝑥𝑦 minus three 𝑥 to the fourth power plus one. (D) 𝑥 squared plus 𝑥 to the power of one-third plus two 𝑥. (E) Negative three 𝑥 plus two 𝑥 plus zero. Which expression, or expressions, is a trinomial?

In this question, we are given a list of five algebraic expressions and asked to determine which of these expressions is a trinomial. We can do this by recalling that for an expression to be a trinomial, it must be both a polynomial and it must have three terms after simplification. This means that we need to check if each of these expressions is a polynomial and if each expression has three terms after simplification.

To do this, we can start by recalling that a polynomial is the sum of monomials, which in turn are the product of constants and variables where the variables must be raised to nonnegative integer exponents. We allow for constant terms in polynomials since raising 𝑥 to an exponent of zero is equivalent to one. We can use this definition to check if each expression is a polynomial by checking if every term in each expression is the product of constants and variables raised to nonnegative integer exponents.

In expression (A), we see that there are two constant terms and only one term containing a variable. We can recall that 𝑥 is the same as 𝑥 to the first power. So the variables in this term are all raised to nonnegative integer exponents. Since every term in this expression is a monomial, we have shown that expression (A) is a polynomial.

We can follow the exact same process for expressions (B), (C), and (E). We can write a variable such as 𝑥 or 𝑦 as 𝑥 to the first power or 𝑦 to the first power and note that every term is the product of constants and variables raised to nonnegative integer exponents. So these are all polynomials.

If we try and apply this process to expression (D), we can note that the variable 𝑥 in the second term has an exponent of one-third. This is not an integer, so this expression is not a polynomial. This in turn also means that this expression cannot be a trinomial.

Now that we have shown that the four remaining expressions are polynomials, we need to determine which expressions have three terms after simplification. It is important to check if we can simplify the expressions. For instance, in expression (A), we can note that negative one plus one is equal to zero. So expression (A) simplifies to give negative 𝑥. Since this expression is a single term and we have shown that it is a polynomial, we can say that this is a monomial, not a trinomial.

In expression (B), we note that there are only two terms. So this is a binomial and not a trinomial. In expression (C), we cannot simplify the expression, and we can see that there are three terms. Since we have shown that this is also a polynomial, it must be a trinomial. Finally, in expression (E), we can note that adding zero will not affect the polynomial. So we can remove this. And then we see that the expression does not have three terms, so it is not a trinomial.

Therefore, of the five given algebraic expressions, we were able to show that only expression (C) is a trinomial.

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