Select The Correct Answer.Which Expression Is Equivalent To This Polynomial Expression?${ \left(2 X^5+3 Y^4\right)\left(-4 X^2+9 Y^4\right) }$A. { -2 X^{10}+11 X^5 Y 4-x 2 Y^4+12 Y^{16}$} B . \[ B. \[ B . \[ -8 X^7+18 X^5 Y^4-12 X^2 Y^4+27
When it comes to multiplying polynomials, it's essential to understand the correct procedure to obtain the equivalent expression. In this article, we will explore the process of multiplying two polynomials and provide a step-by-step guide on how to select the correct answer.
Understanding Polynomial Multiplication
Polynomial multiplication involves multiplying two or more polynomials together. The resulting expression is a polynomial of a higher degree, with each term being the product of the corresponding terms in the original polynomials. To multiply polynomials, we need to follow the distributive property, which states that the product of a sum is equal to the sum of the products.
The Distributive Property
The distributive property is a fundamental concept in algebra that allows us to expand the product of two or more polynomials. It states that for any polynomials P(x) and Q(x), the following equation holds:
P(x) × Q(x) = P(x) × (Q1(x) + Q2(x)) = P(x) × Q1(x) + P(x) × Q2(x)
Multiplying the Given Polynomials
Now, let's apply the distributive property to the given polynomial expression:
(2x^5 + 3y^4) × (-4x^2 + 9y^4)
To multiply these two polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial.
Step 1: Multiply the First Term in the First Polynomial by Each Term in the Second Polynomial
Multiply 2x^5 by -4x^2:
2x^5 × -4x^2 = -8x^7
Multiply 2x^5 by 9y^4:
2x^5 × 9y^4 = 18x5y4
Step 2: Multiply the Second Term in the First Polynomial by Each Term in the Second Polynomial
Multiply 3y^4 by -4x^2:
3y^4 × -4x^2 = -12x2y4
Multiply 3y^4 by 9y^4:
3y^4 × 9y^4 = 27y^8
Step 3: Combine the Terms
Now, let's combine the terms we obtained in the previous steps:
-8x^7 + 18x5y4 - 12x2y4 + 27y^8
Comparing the Result with the Options
Now that we have obtained the equivalent expression, let's compare it with the options provided:
A. -2x^10 + 11x5y4 - x2y4 + 12y^16 B. -8x^7 + 18x5y4 - 12x2y4 + 27y^8
Based on our calculation, the correct answer is:
B. -8x^7 + 18x5y4 - 12x2y4 + 27y^8
Conclusion
In this article, we will address some of the most frequently asked questions on multiplying polynomials. Whether you're a student, teacher, or simply someone looking to brush up on their algebra skills, this article is for you.
Q: What is the distributive property in algebra?
A: The distributive property is a fundamental concept in algebra that allows us to expand the product of two or more polynomials. It states that for any polynomials P(x) and Q(x), the following equation holds:
P(x) × Q(x) = P(x) × (Q1(x) + Q2(x)) = P(x) × Q1(x) + P(x) × Q2(x)
Q: How do I multiply two polynomials?
A: To multiply two polynomials, you need to follow the distributive property. Multiply each term in the first polynomial by each term in the second polynomial, and then combine the terms.
Q: What is the difference between multiplying polynomials and multiplying binomials?
A: Multiplying polynomials involves multiplying two or more polynomials together, while multiplying binomials involves multiplying two binomials together. However, the process is similar, and you can use the distributive property to expand the product.
Q: Can I use the FOIL method to multiply polynomials?
A: No, the FOIL method is used to multiply two binomials together, not polynomials. However, you can use the distributive property to expand the product of two polynomials.
Q: How do I simplify the product of two polynomials?
A: To simplify the product of two polynomials, combine like terms and eliminate any unnecessary terms.
Q: What is the importance of multiplying polynomials in algebra?
A: Multiplying polynomials is an essential skill in algebra, as it allows you to expand the product of two or more polynomials. This is useful in a variety of applications, including solving equations, graphing functions, and modeling real-world problems.
Q: Can I use a calculator to multiply polynomials?
A: Yes, you can use a calculator to multiply polynomials. However, it's essential to understand the process and be able to simplify the product manually.
Q: How do I multiply polynomials with negative exponents?
A: To multiply polynomials with negative exponents, follow the same process as multiplying polynomials with positive exponents. However, be careful when simplifying the product, as negative exponents can lead to complex expressions.
Q: Can I multiply polynomials with fractional exponents?
A: Yes, you can multiply polynomials with fractional exponents. However, be careful when simplifying the product, as fractional exponents can lead to complex expressions.
Conclusion
In this article, we addressed some of the most frequently asked questions on multiplying polynomials. Whether you're a student, teacher, or simply someone looking to brush up on their algebra skills, this article is for you. By understanding the distributive property and following the process of multiplying polynomials, you can simplify complex expressions and solve a variety of problems in algebra.