Multiply:$\[ (2x^2 + 4x - 3)(x^2 - 2x + 5) \\]A. \[$2x^4 + 7x^2 - 15\$\]B. \[$2x^4 - 8x^2 - 15\$\]C. \[$2x^4 - X^2 + 26x - 15\$\]D. \[$2x^4 + 8x^3 - X^2 + 26x - 15\$\]

Introduction

Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with a clear understanding of the process, it becomes a manageable task. In this article, we will explore the step-by-step process of multiplying two polynomials, using the given example: ${(2x^2 + 4x - 3)(x^2 - 2x + 5)}$.

What are 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. Polynomials can be written in the form: $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, and $x$ is the variable.

The Multiplication Process

To multiply two polynomials, we need to follow the distributive property, which states that for any real numbers $a, b, c$, $a(b + c) = ab + ac$. In the context of polynomials, this means that we need to multiply each term in the first polynomial by each term in the second polynomial.

Step 1: Multiply Each Term in the First Polynomial by Each Term in the Second Polynomial

Let's start by multiplying each term in the first polynomial $(2x^2 + 4x - 3)$ by each term in the second polynomial $(x^2 - 2x + 5)$.

• Multiply $2x^2$ by $x^2$: $2x^2 \cdot x^2 = 2x^4$
• Multiply $2x^2$ by $-2x$: $2x^2 \cdot (-2x) = -4x^3$
• Multiply $2x^2$ by $5$: $2x^2 \cdot 5 = 10x^2$
• Multiply $4x$ by $x^2$: $4x \cdot x^2 = 4x^3$
• Multiply $4x$ by $-2x$: $4x \cdot (-2x) = -8x^2$
• Multiply $4x$ by $5$: $4x \cdot 5 = 20x$
• Multiply $-3$ by $x^2$: $-3 \cdot x^2 = -3x^2$
• Multiply $-3$ by $-2x$: $-3 \cdot (-2x) = 6x$
• Multiply $-3$ by $5$: $-3 \cdot 5 = -15$

Step 2: Combine Like Terms

Now that we have multiplied each term in the first polynomial by each term in the second polynomial, we need to combine like terms. Like terms are terms that have the same variable and exponent.

• Combine the terms with $x^4$: $2x^4$
• Combine the terms with $x^3$: $-4x^3 + 4x^3 = 0x^3$ (the terms cancel each other out)
• Combine the terms with $x^2$: $10x^2 - 8x^2 - 3x^2 = -x^2$
• Combine the terms with $x$: $20x + 6x = 26x$
• Combine the constant terms: $-15$

Step 3: Write the Final Answer

Now that we have combined like terms, we can write the final answer.

$\boxed{2x^4 - x^2 + 26x - 15}$

Conclusion

Multiplying polynomials may seem like a daunting task, but with a clear understanding of the process, it becomes a manageable task. By following the distributive property and combining like terms, we can multiply two polynomials and obtain the final answer. In this article, we used the example ${(2x^2 + 4x - 3)(x^2 - 2x + 5)}$ to demonstrate the step-by-step process of multiplying polynomials.

Answer Key

The correct answer is:

$\boxed{2x^4 - x^2 + 26x - 15}$

Q&A: Multiplying Polynomials

In this article, we will continue to explore the concept of multiplying polynomials by answering some frequently asked questions.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a, b, c$, $a(b + c) = ab + ac$. In the context of polynomials, this means that we need to multiply each term in the first polynomial by each term in the second polynomial.

Q: How do I multiply two polynomials?

A: To multiply two polynomials, you need to follow these steps:

1. Multiply each term in the first polynomial by each term in the second polynomial.
2. Combine like terms.
3. Write the final answer.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $-3x^2$ are like terms because they have the same variable ($x$) and exponent ($2$).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have $2x^2$ and $-3x^2$, you can combine them by adding their coefficients: $2x^2 - 3x^2 = -x^2$.

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

A: Multiplying polynomials and multiplying binomials are similar concepts, but they differ in the number of terms involved. Multiplying binomials involves multiplying two binomials (expressions with two terms), while multiplying polynomials involves multiplying two polynomials (expressions with more than two 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 understand the concept and process of multiplying polynomials by hand, as it can help you to better understand the math and to identify any errors.

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 correctly.
• Not following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

Q: How can I practice multiplying polynomials?

A: You can practice multiplying polynomials by working through examples and exercises in your math textbook or online resources. You can also try creating your own examples and challenging yourself to multiply polynomials with different variables and exponents.

Conclusion

Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with practice and patience, it becomes a manageable task. By following the distributive property and combining like terms, we can multiply two polynomials and obtain the final answer. In this article, we answered some frequently asked questions about multiplying polynomials and provided tips and resources for practicing this concept.

Answer Key

The correct answers to the questions are:

• Q: What is the distributive property? A: The distributive property is a mathematical concept that states that for any real numbers $a, b, c$, $a(b + c) = ab + ac$.
• Q: How do I multiply two polynomials? A: To multiply two polynomials, you need to follow these steps: 1. Multiply each term in the first polynomial by each term in the second polynomial. 2. Combine like terms. 3. Write the final answer.
• Q: What are like terms? A: Like terms are terms that have the same variable and exponent.
• Q: How do I combine like terms? A: To combine like terms, you need to add or subtract the coefficients of the like terms.
• Q: What is the difference between multiplying polynomials and multiplying binomials? A: Multiplying polynomials and multiplying binomials are similar concepts, but they differ in the number of terms involved.
• Q: Can I use a calculator to multiply polynomials? A: Yes, you can use a calculator to multiply polynomials.
• 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 correctly. Not following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
• Q: How can I practice multiplying polynomials? A: You can practice multiplying polynomials by working through examples and exercises in your math textbook or online resources. You can also try creating your own examples and challenging yourself to multiply polynomials with different variables and exponents.