Which Is True About The Polynomial { -3xy^2 + 5x^2y$}$?A. It Is A Binomial With A Degree Of 2.B. It Is A Binomial With A Degree Of 3.C. It Is A Trinomial With A Degree Of 2.D. It Is A Trinomial With A Degree Of 3.

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of polynomials and examine the given expression ${-3xy^2 + 5x^2y}$ to determine its characteristics.

What is a Binomial?

A binomial is a polynomial with two terms. It is a simple algebraic expression that consists of two variables or constants combined using addition, subtraction, or multiplication. For example, ${2x + 3y}$ and ${x^2 - 4}$ are binomials. Binomials are an essential part of algebra and are used to solve various mathematical problems.

What is a Trinomial?

A trinomial is a polynomial with three terms. It is a more complex algebraic expression that consists of three variables or constants combined using addition, subtraction, or multiplication. For example, ${2x + 3y - 4}$ and ${x^2 + 4y - 3}$ are trinomials. Trinomials are also an essential part of algebra and are used to solve various mathematical problems.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It is a measure of the complexity of the polynomial. For example, the degree of the polynomial ${x^2 + 3x - 4}$ is 2, while the degree of the polynomial ${x^3 + 2x^2 - 3x + 1}$ is 3.

Analyzing the Given Expression

Now that we have a good understanding of binomials, trinomials, and the degree of a polynomial, let's analyze the given expression ${-3xy^2 + 5x^2y}$. This expression consists of two terms, ${-3xy^2}$ and ${5x^2y}$. Since it has only two terms, it is a binomial.

Determining the Degree of the Polynomial

To determine the degree of the polynomial, we need to find the highest power of the variable in the expression. In this case, the variable is ${x}$ and ${y}$. The highest power of ${x}$ is 2, and the highest power of ${y}$ is 2. Therefore, the degree of the polynomial is 2.

Conclusion

In conclusion, the given expression ${-3xy^2 + 5x^2y}$ is a binomial with a degree of 2. It consists of two terms, and the highest power of the variable is 2. Therefore, option A is the correct answer.

Final Answer

Polynomials are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will address some frequently asked questions about polynomials to provide a deeper understanding of this topic.

Q: What is a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It is a fundamental concept in mathematics and is used to solve various mathematical problems.

Q: What are the different types of polynomials?

There are several types of polynomials, including:

• Monomials: A polynomial with one term, such as ${x^2}$ or ${3y}$.
• Binomials: A polynomial with two terms, such as ${2x + 3y}$ or ${x^2 - 4}$.
• Trinomials: A polynomial with three terms, such as ${2x + 3y - 4}$ or ${x^2 + 4y - 3}$.
• Polynomials with more than three terms: A polynomial with four or more terms, such as ${x^2 + 3x + 2y - 4}$ or ${x^3 + 2x^2 - 3x + 1}$.

Q: What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the expression. It is a measure of the complexity of the polynomial. For example, the degree of the polynomial ${x^2 + 3x - 4}$ is 2, while the degree of the polynomial ${x^3 + 2x^2 - 3x + 1}$ is 3.

Q: How do I add polynomials?

To add polynomials, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, to add the polynomials ${x^2 + 3x - 4}$ and ${2x^2 + 4x + 3}$, you need to combine the like terms:

• Combine the ${x^2}$ terms: ${x^2 + 2x^2 = 3x^2}$
• Combine the ${x}$ terms: ${3x + 4x = 7x}$
• Combine the constant terms: ${-4 + 3 = -1}$

Therefore, the sum of the two polynomials is ${3x^2 + 7x - 1}$.

Q: How do I subtract polynomials?

To subtract polynomials, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, to subtract the polynomials ${x^2 + 3x - 4}$ and ${2x^2 + 4x + 3}$, you need to combine the like terms:

• Combine the ${x^2}$ terms: ${x^2 - 2x^2 = -x^2}$
• Combine the ${x}$ terms: ${3x - 4x = -x}$
• Combine the constant terms: ${-4 - 3 = -7}$

Therefore, the difference of the two polynomials is ${-x^2 - x - 7}$.

Q: How do I multiply polynomials?

To multiply polynomials, you need to use the distributive property. The distributive property states that for any polynomials ${a}$ and ${b}$, and any variable ${x}$, ${a(bx) = abx}$. For example, to multiply the polynomials ${x^2 + 3x - 4}$ and ${2x + 4}$, you need to use the distributive property:

• Multiply the first term of the first polynomial by the entire second polynomial: ${x^2(2x + 4) = 2x^3 + 4x^2}$
• Multiply the second term of the first polynomial by the entire second polynomial: ${3x(2x + 4) = 6x^2 + 12x}$
• Multiply the third term of the first polynomial by the entire second polynomial: ${-4(2x + 4) = -8x - 16}$

Therefore, the product of the two polynomials is ${2x^3 + 10x^2 + 12x - 16}$.

Q: How do I divide polynomials?

To divide polynomials, you need to use long division. Long division is a method of dividing polynomials by dividing the leading term of the dividend by the leading term of the divisor. For example, to divide the polynomial ${x^3 + 2x^2 - 3x + 1}$ by the polynomial ${x + 2}$, you need to use long division:

• Divide the leading term of the dividend by the leading term of the divisor: ${x^3 / (x + 2) = x^2}$
• Multiply the divisor by the result: ${(x + 2)x^2 = x^3 + 2x^2}$
• Subtract the result from the dividend: ${x^3 + 2x^2 - 3x + 1 - (x^3 + 2x^2) = -3x + 1}$
• Repeat the process with the new dividend: ${-3x + 1 / (x + 2) = -3}$

Therefore, the quotient of the two polynomials is ${x^2 - 3}$.

Conclusion

In conclusion, polynomials are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By understanding how to add, subtract, multiply, and divide polynomials, you can solve a wide range of mathematical problems.