Find The Product: \left(\frac{1}{2} X^2 - 4\right)\left(\frac{1}{4} X^3 - 3x + 2\right ]A. 1 8 X 6 − X 3 − 1 2 X 2 + 12 X − 8 \frac{1}{8} X^6 - X^3 - \frac{1}{2} X^2 + 12x - 8 8 1 ​ X 6 − X 3 − 2 1 ​ X 2 + 12 X − 8 B. 1 8 X 5 − 5 2 X 3 + X 2 + 12 X − 8 \frac{1}{8} X^5 - \frac{5}{2} X^3 + X^2 + 12x - 8 8 1 ​ X 5 − 2 5 ​ X 3 + X 2 + 12 X − 8 C. $\frac{3}{4} X^5 -

Introduction

Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with a clear understanding of the steps involved, it becomes a manageable task. In this article, we will guide you through the process of multiplying two polynomials, using the given example to illustrate the concept.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in the form:

a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and x is the variable.

The Product of Two Polynomials

To find the product of two polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial. This can be done using the distributive property, which states that:

a(b + c) = ab + ac

Using this property, we can multiply each term in the first polynomial by each term in the second polynomial.

Example: Multiplying Two Polynomials

Let's consider the given example:

$\left(\frac{1}{2} x^2 - 4\right)\left(\frac{1}{4} x^3 - 3x + 2\right)$

To find the product, 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

The first term in the first polynomial is $\frac{1}{2} x^2$. We need to multiply this term by each term in the second polynomial:

$\frac{1}{2} x^2 \cdot \frac{1}{4} x^3 = \frac{1}{8} x^5$

$\frac{1}{2} x^2 \cdot (-3x) = -\frac{3}{2} x^3$

$\frac{1}{2} x^2 \cdot 2 = x^2$

Step 2: Multiply the Second Term in the First Polynomial by Each Term in the Second Polynomial

The second term in the first polynomial is $-4$. We need to multiply this term by each term in the second polynomial:

$-4 \cdot \frac{1}{4} x^3 = -x^3$

$-4 \cdot (-3x) = 12x$

$-4 \cdot 2 = -8$

Step 3: Combine the Terms

Now that we have multiplied each term in the first polynomial by each term in the second polynomial, we need to combine the terms. We can do this by adding or subtracting the terms as necessary.

The product of the two polynomials is:

$\frac{1}{8} x^5 - \frac{3}{2} x^3 + x^2 - x^3 + 12x - 8$

Combining like terms, we get:

$\frac{1}{8} x^5 - \frac{5}{2} x^3 + x^2 + 12x - 8$

Conclusion

Multiplying polynomials is a straightforward process that involves multiplying each term in the first polynomial by each term in the second polynomial and combining the terms. By following the steps outlined in this article, you should be able to find the product of two polynomials with ease.

Answer

The correct answer is:

$\boxed{\frac{1}{8} x^5 - \frac{5}{2} x^3 + x^2 + 12x - 8}$

Discussion

Do you have any questions about multiplying polynomials? Have you tried multiplying polynomials before? Share your experiences and tips in the comments below!

Related Topics

• Adding and Subtracting Polynomials
• Multiplying and Dividing Polynomials
• Factoring Polynomials
• Quadratic Equations

Resources

• Khan Academy: Multiplying Polynomials
• Mathway: Multiplying Polynomials
• Wolfram Alpha: Multiplying Polynomials

Final Thoughts

Introduction

Multiplying polynomials can be a challenging task, but with the right guidance, it can be made easier. In this article, we will answer some of the most frequently asked questions about multiplying polynomials, providing you with a better understanding of the concept.

Q: What is the difference between multiplying polynomials and multiplying numbers?

A: Multiplying polynomials is similar to multiplying numbers, but with the added complexity of variables and exponents. When multiplying polynomials, you need to multiply each term in the first polynomial by each term in the second polynomial, taking into account the exponents and coefficients.

Q: How do I multiply polynomials with different exponents?

A: When multiplying polynomials with different exponents, you need to multiply the coefficients and add the exponents. For example, if you have the polynomials x^2 and x^3, the product would be x^(2+3) = x^5.

Q: What is the distributive property, and how is it used in multiplying polynomials?

A: The distributive property is a mathematical concept that states that a(b + c) = ab + ac. In the context of multiplying polynomials, the distributive property is used to multiply each term in the first polynomial by each term in the second polynomial.

Q: How do I handle negative coefficients when multiplying polynomials?

A: When multiplying polynomials with negative coefficients, you need to multiply the coefficients and keep the sign. For example, if you have the polynomials -x and -y, the product would be x*y = -xy.

Q: Can I use a calculator to multiply polynomials?

A: Yes, you can use a calculator to multiply polynomials, but it's essential to understand the concept behind the calculation. Using a calculator can help you verify your work and ensure that you have the correct answer.

Q: How do I simplify the product of two polynomials?

A: To simplify the product of two polynomials, you need to combine like terms and eliminate any unnecessary terms. This involves adding or subtracting the coefficients of the same variables.

Q: What are some common mistakes to avoid when multiplying polynomials?

A: Some common mistakes to avoid when multiplying polynomials include:

• Forgetting to multiply each term in the first polynomial by each term in the second polynomial
• Not combining like terms
• Not handling negative coefficients correctly
• Not simplifying the product

Q: Can I use the FOIL method to multiply polynomials?

A: Yes, you can use the FOIL method to multiply polynomials, but it's essential to understand the concept behind the method. The FOIL method involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: How do I multiply polynomials with multiple variables?

A: When multiplying polynomials with multiple variables, you need to multiply each term in the first polynomial by each term in the second polynomial, taking into account the exponents and coefficients of each variable.

Conclusion

Multiplying polynomials can be a challenging task, but with the right guidance, it can be made easier. By understanding the concept behind multiplying polynomials and following the steps outlined in this article, you should be able to multiply polynomials with ease.

Related Topics

• Adding and Subtracting Polynomials
• Multiplying and Dividing Polynomials
• Factoring Polynomials
• Quadratic Equations

Resources

• Khan Academy: Multiplying Polynomials
• Mathway: Multiplying Polynomials
• Wolfram Alpha: Multiplying Polynomials

Final Thoughts

Multiplying polynomials is an essential skill in algebra that can be used to solve a wide range of problems. By understanding the concept behind multiplying polynomials and following the steps outlined in this article, you should be able to multiply polynomials with ease. Remember to always combine like terms and check your work to ensure that you have the correct answer. Happy calculating!