2. Using The Expressions \[$\left(-3x^3 + 2x - 5\right)\$\] And \[$\left(2x^4 - 4x^2 - 3\right)\$\]:(a) Calculate The Product Of The Polynomials. Show Your Work And Provide The Answer In Standard Form.(b) Is The Product Of

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Introduction

In algebra, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we multiply polynomials, we are essentially combining these expressions to form a new expression. In this article, we will explore how to multiply two polynomials using the given expressions {\left(-3x^3 + 2x - 5\right)$}$ and {\left(2x^4 - 4x^2 - 3\right)$}$.

Understanding the Problem

To multiply these polynomials, we need to apply the distributive property, which states that for any real numbers a, b, and c:

a(b + c) = ab + ac

We will use this property 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

We will start by multiplying the first term in the first polynomial, −3x3{-3x^3}, by each term in the second polynomial.

Multiply −3x3{-3x^3} by 2x4{2x^4}

−3x3⋅2x4=−6x7{-3x^3 \cdot 2x^4 = -6x^7}

Multiply −3x3{-3x^3} by −4x2{-4x^2}

−3x3⋅−4x2=12x5{-3x^3 \cdot -4x^2 = 12x^5}

Multiply −3x3{-3x^3} by −3{-3}

−3x3⋅−3=9x3{-3x^3 \cdot -3 = 9x^3}

Multiply 2x{2x} by 2x4{2x^4}

2xâ‹…2x4=4x5{2x \cdot 2x^4 = 4x^5}

Multiply 2x{2x} by −4x2{-4x^2}

2x⋅−4x2=−8x3{2x \cdot -4x^2 = -8x^3}

Multiply 2x{2x} by −3{-3}

2x⋅−3=−6x{2x \cdot -3 = -6x}

Multiply −5{-5} by 2x4{2x^4}

−5⋅2x4=−10x4{-5 \cdot 2x^4 = -10x^4}

Multiply −5{-5} by −4x2{-4x^2}

−5⋅−4x2=20x2{-5 \cdot -4x^2 = 20x^2}

Multiply −5{-5} by −3{-3}

−5⋅−3=15{-5 \cdot -3 = 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 the same exponent

x7{x^7} term: −6x7{-6x^7}

x5{x^5} terms: 12x5+4x5=16x5{12x^5 + 4x^5 = 16x^5}

x4{x^4} term: −10x4{-10x^4}

x3{x^3} terms: 9x3−8x3=x3{9x^3 - 8x^3 = x^3}

x2{x^2} term: 20x2{20x^2}

x{x} term: −6x{-6x}

Constant term: 15{15}

The Final Answer

The product of the polynomials is:

−6x7+16x5−10x4+x3+20x2−6x+15{-6x^7 + 16x^5 - 10x^4 + x^3 + 20x^2 - 6x + 15}

Conclusion

Multiplying polynomials can be a challenging task, but by applying the distributive property and combining like terms, we can simplify the process. In this article, we have shown how to multiply two polynomials using the given expressions {\left(-3x^3 + 2x - 5\right)$}$ and {\left(2x^4 - 4x^2 - 3\right)$}$. We have also provided the final answer in standard form.

Is the Product of the Polynomials a Polynomial?

Yes, the product of the polynomials is a polynomial. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The product of the polynomials meets this definition, as it is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Why is it Important to Multiply Polynomials?

Multiplying polynomials is an important concept in algebra, as it allows us to simplify complex expressions and solve equations. By multiplying polynomials, we can:

  • Simplify complex expressions
  • Solve equations
  • Graph functions
  • Find the roots of a polynomial

Introduction

Multiplying polynomials can be a challenging task, but with the right techniques and understanding, it can be simplified. In this article, we will provide a Q&A guide to help you understand the concept of multiplying polynomials.

Q: What is a polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

A: A polynomial can be written in the form:

anxn+an−1xn−1+…+a1x+a0{a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0}

where an,an−1,…,a1,a0{a_n, a_{n-1}, \ldots, a_1, a_0} are coefficients, and x{x} is the variable.

Q: What is the distributive property?

The distributive property is a mathematical concept that states that for any real numbers a, b, and c:

a(b + c) = ab + ac

A: The distributive property is used to multiply each term in the first polynomial by each term in the second polynomial.

Q: How do I multiply polynomials?

To multiply polynomials, you need to apply the distributive property and combine like terms.

A: Here's a step-by-step guide to multiplying polynomials:

  1. Multiply each term in the first polynomial by each term in the second polynomial.
  2. Combine like terms.

Q: What are like terms?

Like terms are terms that have the same variable and exponent.

A: For example, in the expression 2x2+3x2{2x^2 + 3x^2}, the terms 2x2{2x^2} and 3x2{3x^2} are like terms because they have the same variable and exponent.

Q: How do I combine like terms?

To combine like terms, you need to add or subtract the coefficients of the like terms.

A: For example, in the expression 2x2+3x2{2x^2 + 3x^2}, you can combine the like terms by adding the coefficients:

2x2+3x2=5x2{2x^2 + 3x^2 = 5x^2}

Q: What is the product of two polynomials?

The product of two polynomials is a new polynomial that is formed by multiplying each term in the first polynomial by each term in the second polynomial.

A: For example, if we multiply the polynomials x2+2x+1{x^2 + 2x + 1} and x+1{x + 1}, the product is:

x3+3x2+2x+1{x^3 + 3x^2 + 2x + 1}

Q: Why is it important to multiply polynomials?

Multiplying polynomials is an important concept in algebra because it allows us to simplify complex expressions and solve equations.

A: By multiplying polynomials, we can:

  • Simplify complex expressions
  • Solve equations
  • Graph functions
  • Find the roots of a polynomial

Conclusion

Multiplying polynomials can be a challenging task, but with the right techniques and understanding, it can be simplified. In this article, we have provided a Q&A guide to help you understand the concept of multiplying polynomials. We hope that this guide has been helpful in answering your questions and providing a better understanding of the concept.

Frequently Asked Questions

  • Q: What is the difference between a polynomial and a rational expression?
  • A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression is a fraction of two polynomials.
  • Q: How do I multiply a polynomial by a rational expression?
  • A: To multiply a polynomial by a rational expression, you need to multiply the polynomial by the numerator of the rational expression and then divide the result by the denominator of the rational expression.
  • Q: What is the product of a polynomial and a constant?
  • A: The product of a polynomial and a constant is a new polynomial that is formed by multiplying each term in the polynomial by the constant.

Additional Resources

  • Polynomial Multiplication Worksheet: A worksheet that provides practice problems for multiplying polynomials.
  • Polynomial Multiplication Video: A video that provides a step-by-step guide to multiplying polynomials.
  • Polynomial Multiplication Tutorial: A tutorial that provides a detailed explanation of the concept of multiplying polynomials.