Determine which of the following are trinomials when simplified.
9. $x-3 y+z$
10. $11 a-a+8$
11. $y^2-3 y^2$
12. $a b c+d-7 a b c+9 d$
13. $4 f g-3 d^2+e^5$
14. $x^5 y^7 z^9$
15. $4 x-3 z$
16. $21 b-b+5 b$
17. $5 k^2-13 m^2$
18. $\frac{1}{2} h g-\frac{3}{4} h g$
19. $14 w x-6 z^2+y^5$
20. -87
21. $\frac{x}{2}+\frac{x}{4}$
22. $x^3-2 x^2+4 x^3-2$
Answer (1)
Simplify each expression by combining like terms.
Count the number of terms in the simplified expression.
Identify expressions with exactly three terms as trinomials.
The trinomials are x − 3 y + z , 4 f g − 3 d 2 + e 5 , 14 w x − 6 z 2 + y 5 , and 5 x 3 − 2 x 2 − 2 .
Explanation
Understanding the Problem We need to identify which of the given expressions, when simplified, are trinomials. A trinomial is a polynomial expression with exactly three terms. We will simplify each expression and count the number of terms in the simplified expression. If the simplified expression has exactly three terms, then it is a trinomial.
Simplifying the Expressions
x − 3 y + z : This expression already has three terms. It is a trinomial.
11 a − a + 8 : Simplifying, we get 10 a + 8 . This expression has two terms. It is a binomial.
y 2 − 3 y 2 : Simplifying, we get − 2 y 2 . This expression has one term. It is a monomial.
ab c + d − 7 ab c + 9 d : Simplifying, we get − 6 ab c + 10 d . This expression has two terms. It is a binomial.
4 f g − 3 d 2 + e 5 : This expression already has three terms. It is a trinomial.
x 5 y 7 z 9 : This expression has one term. It is a monomial.
4 x − 3 z : This expression already has two terms. It is a binomial.
21 b − b + 5 b : Simplifying, we get 25 b . This expression has one term. It is a monomial.
5 k 2 − 13 m 2 : This expression already has two terms. It is a binomial.
2 1 h g − 4 3 h g : Simplifying, we get − 4 1 h g . This expression has one term. It is a monomial.
14 w x − 6 z 2 + y 5 : This expression already has three terms. It is a trinomial.
-87: This expression has one term. It is a monomial.
2 x + 4 x : Simplifying, we get 4 3 x . This expression has one term. It is a monomial.
x 3 − 2 x 2 + 4 x 3 − 2 : Simplifying, we get 5 x 3 − 2 x 2 − 2 . This expression has three terms. It is a trinomial.
Identifying Trinomials The trinomials are:
x − 3 y + z
4 f g − 3 d 2 + e 5
14 w x − 6 z 2 + y 5
x 3 − 2 x 2 + 4 x 3 − 2 which simplifies to 5 x 3 − 2 x 2 − 2 .
Final Answer The expressions that simplify to trinomials are: x − 3 y + z , 4 f g − 3 d 2 + e 5 , 14 w x − 6 z 2 + y 5 , and 5 x 3 − 2 x 2 − 2 .
Examples
Trinomials are useful in various fields, such as physics and engineering, where they can represent complex relationships between variables. For example, the equation of motion of a projectile under constant acceleration can be expressed as a trinomial in terms of time. Understanding trinomials helps in solving problems related to projectile motion, such as determining the time of flight or the maximum height reached. Similarly, in electrical engineering, trinomials can be used to model the behavior of circuits with multiple components.
