Finding the Difference of Two Polynomials

Oct 13, 2025 by leolavanderia.com.br 85 views

Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we are given two polynomials, we can find their difference by subtracting one from the other. In this article, we will explore how to find the difference of two polynomials and apply this concept to a specific problem.

What are Polynomials?

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They can be written in the form of:

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.

Finding the Difference of Two Polynomials

To find the difference of two polynomials, we simply subtract one from the other. This means that we will combine like terms and simplify the resulting expression.

For example, let's consider two polynomials:

p(x) = 2x^2 + 3x - 1

q(x) = x^2 + 2x + 1

To find the difference of these two polynomials, we will subtract $q(x)$ from $p(x)$ :

p(x) - q(x) = (2x^2 + 3x - 1) - (x^2 + 2x + 1)

= 2x^2 + 3x - 1 - x^2 - 2x - 1

= x^2 + x - 2

Applying the Concept to a Specific Problem

Now, let's apply the concept of finding the difference of two polynomials to the problem given in the question:

\begin{aligned} (12x^2 - 11y^2 - 13x) - (5x^2 - 14y^2 - 9x) \end{aligned}

To find the difference of these two polynomials, we will subtract the second polynomial from the first polynomial:

\begin{aligned} (12x^2 - 11y^2 - 13x) - (5x^2 - 14y^2 - 9x) \end{aligned}

= 12x^2 - 11y^2 - 13x - 5x^2 + 14y^2 + 9x

= 7x^2 + 3y^2 - 4x

Conclusion

In this article, we explored how to find the difference of two polynomials and applied this concept to a specific problem. We saw that the difference of two polynomials can be found by subtracting one from the other and combining like terms. By following this process, we can find the difference of any two polynomials.

Final Answer

The final answer to the problem is:

\boxed{7x^2 + 3y^2 - 4x}

This answer matches option A in the given problem.

Introduction

In our previous article, we explored the concept of polynomials and how to find the difference of two polynomials. In this article, we will answer some frequently asked questions (FAQs) on polynomials to help you better understand this topic.

Q1: What is a polynomial?

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

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.

Q2: What are the different types of polynomials?

There are several types of polynomials, including:

Monomial: A polynomial with only one term, such as $x^2$ .
Binomial: A polynomial with two terms, such as $x^2 + 3x$ .
Trinomial: A polynomial with three terms, such as $x^2 + 3x + 2$ .
Quadratic: A polynomial of degree two, such as $x^2 + 3x + 2$ .
Cubic: A polynomial of degree three, such as $x^3 + 3x^2 + 2x + 1$ .

Q3: How do I add polynomials?

To add polynomials, we simply combine like terms. For example, let's consider two polynomials:

p(x) = 2x^2 + 3x - 1

q(x) = x^2 + 2x + 1

To add these two polynomials, we will combine like terms:

p(x) + q(x) = (2x^2 + 3x - 1) + (x^2 + 2x + 1)

= 2x^2 + 3x - 1 + x^2 + 2x + 1

= 3x^2 + 5x

Q4: How do I subtract polynomials?

To subtract polynomials, we simply subtract one from the other. For example, let's consider two polynomials:

p(x) = 2x^2 + 3x - 1

q(x) = x^2 + 2x + 1

To subtract $q(x)$ from $p(x)$ , we will subtract one from the other:

p(x) - q(x) = (2x^2 + 3x - 1) - (x^2 + 2x + 1)

= 2x^2 + 3x - 1 - x^2 - 2x - 1

= x^2 + x - 2

Q5: How do I multiply polynomials?

To multiply polynomials, we use the distributive property. For example, let's consider two polynomials:

p(x) = 2x^2 + 3x - 1

q(x) = x^2 + 2x + 1

To multiply these two polynomials, we will use the distributive property:

p(x) \cdot q(x) = (2x^2 + 3x - 1) \cdot (x^2 + 2x + 1)

= 2x^2 \cdot x^2 + 2x^2 \cdot 2x + 2x^2 \cdot 1 + 3x \cdot x^2 + 3x \cdot 2x + 3x \cdot 1 - 1 \cdot x^2 - 1 \cdot 2x - 1 \cdot 1

= 2x^4 + 4x^3 + 2x^2 + 3x^3 + 6x^2 + 3x - x^2 - 2x - 1

= 2x^4 + 7x^3 + 7x^2 + x - 1

Q6: What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $x^2 + 3x + 2$ , the degree is 2.

Q7: How do I factor a polynomial?

Factoring a polynomial involves expressing it as a product of simpler polynomials. For example, let's consider the polynomial $x^2 + 3x + 2$ . We can factor this polynomial as:

x^2 + 3x + 2 = (x + 1)(x + 2)

Conclusion

In this article, we answered some frequently asked questions (FAQs) on polynomials to help you better understand this topic. We covered topics such as the definition of a polynomial, the different types of polynomials, how to add and subtract polynomials, how to multiply polynomials, the degree of a polynomial, and how to factor a polynomial.

Final Answer

We hope this article has been helpful in answering your questions about polynomials. If you have any further questions, please don't hesitate to ask.