Drag the Operator to the Correct Location

Oct 13, 2025 by miodeya.com 120 views

Introduction

In mathematics, a binomial is a polynomial with two terms. It is a fundamental concept in algebra and is used extensively in various mathematical operations. When dealing with binomials, it is essential to understand the different operations that can be performed on them. In this article, we will discuss the operation that results in a binomial and provide examples to illustrate the concept.

What is a Binomial?

A binomial is a polynomial with two terms. It can be written in the form of:

ax + b

where a and b are constants, and x is the variable.

Examples of Binomials

2x + 3
x^2 + 4
3y^6 + 4

Operations on Binomials

There are several operations that can be performed on binomials, including:

Addition: Adding two binomials results in a new binomial.
Subtraction: Subtracting one binomial from another results in a new binomial.
Multiplication: Multiplying two binomials results in a new polynomial.
Division: Dividing one binomial by another results in a new polynomial.

Which Operation Results in a Binomial?

To determine which operation results in a binomial, we need to examine the given expression:

$\left(3y^6 + 4\right) \quad \left(9y^{12} - 12y^6 + 16\right)$

In this expression, we have two binomials multiplied together. When we multiply two binomials, the result is a new polynomial. However, if we multiply two binomials and the resulting expression has only two terms, then it is a binomial.

Step 1: Multiply the Two Binomials

To multiply the two binomials, we need to multiply each term in the first binomial by each term in the second binomial.

$\left(3y^6 + 4\right) \quad \left(9y^{12} - 12y^6 + 16\right)$

$= 3y^6 \times 9y^{12} - 3y^6 \times 12y^6 + 3y^6 \times 16 + 4 \times 9y^{12} - 4 \times 12y^6 + 4 \times 16$

$= 27y^{18} - 36y^{12} + 48y^6 + 36y^{12} - 48y^6 + 64$

Step 2: Simplify the Expression

Now, we need to simplify the expression by combining like terms.

$= 27y^{18} - 36y^{12} + 48y^6 + 36y^{12} - 48y^6 + 64$

$= 27y^{18} + 0y^{12} + 0y^6 + 64$

$= 27y^{18} + 64$

Conclusion

In conclusion, when we multiply two binomials, the result is a new polynomial. However, if we multiply two binomials and the resulting expression has only two terms, then it is a binomial. In this case, the resulting expression is a binomial because it has only two terms.

Drag the Operator to the Correct Location

To determine which operation results in a binomial, we need to examine the given expression:

$\left(3y^6 + 4\right) \quad \left(9y^{12} - 12y^6 + 16\right)$

The correct operation is multiplication. The resulting expression is a binomial because it has only two terms.

Final Answer

Q&A: Binomial Operations

Q: What is a binomial?

A: A binomial is a polynomial with two terms. It can be written in the form of:

ax + b

where a and b are constants, and x is the variable.

Q: What are some examples of binomials?

A: Some examples of binomials include:

2x + 3
x^2 + 4
3y^6 + 4

Q: What are the different operations that can be performed on binomials?

A: There are several operations that can be performed on binomials, including:

Addition: Adding two binomials results in a new binomial.
Subtraction: Subtracting one binomial from another results in a new binomial.
Multiplication: Multiplying two binomials results in a new polynomial.
Division: Dividing one binomial by another results in a new polynomial.

Q: Which operation results in a binomial?

A: To determine which operation results in a binomial, we need to examine the given expression:

$\left(3y^6 + 4\right) \quad \left(9y^{12} - 12y^6 + 16\right)$

Q: How do we multiply two binomials?

A: To multiply two binomials, we need to multiply each term in the first binomial by each term in the second binomial.

$\left(3y^6 + 4\right) \quad \left(9y^{12} - 12y^6 + 16\right)$

$= 3y^6 \times 9y^{12} - 3y^6 \times 12y^6 + 3y^6 \times 16 + 4 \times 9y^{12} - 4 \times 12y^6 + 4 \times 16$

$= 27y^{18} - 36y^{12} + 48y^6 + 36y^{12} - 48y^6 + 64$

Q: How do we simplify the expression?

A: Now, we need to simplify the expression by combining like terms.

$= 27y^{18} - 36y^{12} + 48y^6 + 36y^{12} - 48y^6 + 64$

$= 27y^{18} + 0y^{12} + 0y^6 + 64$

$= 27y^{18} + 64$

Q: What is the final answer?

A: The final answer is multiplication. The resulting expression is a binomial because it has only two terms.

Common Mistakes to Avoid

Not following the order of operations: When multiplying two binomials, it is essential to follow the order of operations (PEMDAS) to ensure that the correct result is obtained.
Not combining like terms: When simplifying the expression, it is crucial to combine like terms to ensure that the correct result is obtained.
Not checking the final answer: Before providing the final answer, it is essential to check that the resulting expression has only two terms.

Conclusion

In conclusion, understanding the correct location of the operator is crucial when dealing with binomial operations. By following the order of operations and combining like terms, we can ensure that the correct result is obtained.