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Mar 13, 2025 by ADMIN 260 views

**Multiplying Polynomials: A Step-by-Step Guide**

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Introduction

Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with a step-by-step approach, it becomes manageable. In this article, we will explore how to multiply two polynomials, focusing on the given problem: $\left(4x^2 + 2\right)\left(6x^2 + 8x + 5\right)$ . We will break down the process into manageable steps, making it easier to understand and apply.

Understanding Polynomials

Before we dive into the multiplication process, let's briefly discuss what polynomials are. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as $x$ , $y$ , or $z$ . The coefficients are the numerical values that are multiplied with the variables.

The FOIL Method

The FOIL method is a popular technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which we multiply the terms. However, in this case, we are multiplying two polynomials, not binomials. We will use a modified version of the FOIL method to multiply the given polynomials.

Step 1: Multiply the First Terms

To multiply the first terms, we multiply the first term of the first polynomial by the first term of the second polynomial. In this case, we have $4x^2$ and $6x^2$ . Multiplying these two terms gives us $24x^4$ .

Step 2: Multiply the Outer Terms

Next, we multiply the first term of the first polynomial by the last term of the second polynomial. In this case, we have $4x^2$ and $5$ . Multiplying these two terms gives us $20x^2$ .

Step 3: Multiply the Inner Terms

Now, we multiply the last term of the first polynomial by the first term of the second polynomial. In this case, we have $2$ and $6x^2$ . Multiplying these two terms gives us $12x^2$ .

Step 4: Multiply the Last Terms

Finally, we multiply the last term of the first polynomial by the last term of the second polynomial. In this case, we have $2$ and $5$ . Multiplying these two terms gives us $10$ .

Combining the Terms

Now that we have multiplied all the terms, we need to combine them. We add the terms with the same variable and exponent. In this case, we have $24x^4$ , $20x^2$ , and $12x^2$ . Combining these terms gives us $24x^4 + 32x^2 + 10$ .

Conclusion

In conclusion, multiplying polynomials can seem daunting at first, but with a step-by-step approach, it becomes manageable. By following the modified FOIL method, we can multiply two polynomials and combine the terms to get the final result. In this case, the product of $\left(4x^2 + 2\right)\left(6x^2 + 8x + 5\right)$ is $24x^4 + 32x^2 + 10$ .

Final Answer

$\boxed{24x^4 + 32x^2 + 10}$

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Q&A: Multiplying Polynomials

Q: What is the FOIL method?

A: The FOIL method is a technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which we multiply the terms. However, in this case, we are multiplying two polynomials, not binomials. We will use a modified version of the FOIL method to multiply the given polynomials.

Q: How do I multiply the first terms?

A: To multiply the first terms, we multiply the first term of the first polynomial by the first term of the second polynomial. In this case, we have $4x^2$ and $6x^2$ . Multiplying these two terms gives us $24x^4$ .

Q: What about the outer terms? How do I multiply them?

A: Next, we multiply the first term of the first polynomial by the last term of the second polynomial. In this case, we have $4x^2$ and $5$ . Multiplying these two terms gives us $20x^2$ .

Q: How do I multiply the inner terms?

A: Now, we multiply the last term of the first polynomial by the first term of the second polynomial. In this case, we have $2$ and $6x^2$ . Multiplying these two terms gives us $12x^2$ .

Q: What about the last terms? How do I multiply them?

A: Finally, we multiply the last term of the first polynomial by the last term of the second polynomial. In this case, we have $2$ and $5$ . Multiplying these two terms gives us $10$ .

Q: How do I combine the terms?

A: Now that we have multiplied all the terms, we need to combine them. We add the terms with the same variable and exponent. In this case, we have $24x^4$ , $20x^2$ , and $12x^2$ . Combining these terms gives us $24x^4 + 32x^2 + 10$ .

Q: What if I have a polynomial with more than two terms? How do I multiply it?

A: If you have a polynomial with more than two terms, you can use the distributive property to multiply each term by each other term. For example, if you have the polynomial $3x^2 + 2x + 1$ and you want to multiply it by $2x^2 + 3x + 4$ , you can multiply each term of the first polynomial by each term of the second polynomial and then combine the terms.

Q: Can I use a calculator to multiply polynomials?

A: Yes, you can use a calculator to multiply polynomials. However, it's always a good idea to check your work by multiplying the polynomials by hand to make sure you get the correct answer.

Q: What if I make a mistake when multiplying polynomials? How do I correct it?

A: If you make a mistake when multiplying polynomials, don't worry! It's easy to correct. Just go back and recheck your work, making sure to multiply each term correctly. If you're still having trouble, try using a different method, such as the distributive property, to multiply the polynomials.

Conclusion

Multiplying polynomials can seem daunting at first, but with a step-by-step approach, it becomes manageable. By following the modified FOIL method and using the distributive property, you can multiply polynomials and combine the terms to get the final result. Remember to always check your work and use a calculator if needed. With practice, you'll become a pro at multiplying polynomials in no time!

Final Answer

$\boxed{24x^4 + 32x^2 + 10}$