Which Expression Can Be Used To Find The Difference Of The Polynomials?A. { (10m - 6) - (7m - 4)$}$B. { [10m + (-7m)] + [(-6) + 4]$}$C. { (10m + 7m) + [(-6) + (-4)]$}$D. { [(-10m) + (-7m)] + (6 + 4)$}$E.
Introduction
In algebra, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we need to find the difference of two polynomials, we can use various methods to simplify the expression. In this article, we will explore the correct method to find the difference of polynomials and evaluate the given options.
What is the Difference of Polynomials?
The difference of two polynomials is the result of subtracting one polynomial from another. For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2)
Evaluating the Options
Now, let's evaluate the given options to find the correct expression for the difference of polynomials.
Option A: {(10m - 6) - (7m - 4)$}$
To find the difference, we need to subtract the second polynomial from the first. We can start by distributing the negative sign to the terms inside the second polynomial:
{(10m - 6) - (7m - 4)$ = 10m - 6 - 7m + 4
Combine like terms:
[$10m - 7m - 6 + 4$ = 3m - 2
This is a valid expression for the difference of polynomials.
Option B: [[10m + (-7m)] + [(-6) + 4]\$}
This option involves combining like terms and adding the results. However, this method is not the most efficient way to find the difference of polynomials.
{[10m + (-7m)] + [(-6) + 4]$ = 3m - 2
This expression is equivalent to Option A, but it's not the most straightforward way to find the difference.
Option C: [(10m + 7m) + [(-6) + (-4)]\$}
This option involves combining like terms and adding the results. However, this method is not the most efficient way to find the difference of polynomials.
{(10m + 7m) + [(-6) + (-4)]$ = 17m - 10
This expression is not equivalent to the correct difference.
Option D: [[(-10m) + (-7m)] + (6 + 4)\$}
This option involves combining like terms and adding the results. However, this method is not the most efficient way to find the difference of polynomials.
{[(-10m) + (-7m)] + (6 + 4)$ = -17m + 10
This expression is not equivalent to the correct difference.
Option E: [$(10m - 6) - (7m - 4)$]
This option is equivalent to Option A, which is a valid expression for the difference of polynomials.
Conclusion
In conclusion, the correct expression for finding the difference of polynomials is Option A: [(10m - 6) - (7m - 4)\$}. This method involves distributing the negative sign to the terms inside the second polynomial and combining like terms. The other options, while equivalent to the correct expression, are not the most efficient way to find the difference of polynomials.
Final Answer
Q: What is the difference of polynomials?
A: The difference of two polynomials is the result of subtracting one polynomial from another. For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2)
Q: How do I find the difference of polynomials?
A: To find the difference of polynomials, you can use the following steps:
- Distribute the negative sign to the terms inside the second polynomial.
- Combine like terms.
- Simplify the expression.
For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2) = 3x^2 + 2x - 1 - 2x^2 + 3x - 2 = x^2 + 5x - 3
Q: What is the difference between the difference of polynomials and the sum of polynomials?
A: The difference of polynomials is the result of subtracting one polynomial from another, while the sum of polynomials is the result of adding two or more polynomials together.
For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The sum of p(x) and q(x) is:
p(x) + q(x) = (3x^2 + 2x - 1) + (2x^2 - 3x + 2) = 5x^2 - x + 1
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2) = x^2 + 5x - 3
Q: Can I use the distributive property to find the difference of polynomials?
A: Yes, you can use the distributive property to find the difference of polynomials. The distributive property states that for any real numbers a, b, and c:
a(b + c) = ab + ac
You can use this property to distribute the negative sign to the terms inside the second polynomial.
For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2) = 3x^2 + 2x - 1 - 2x^2 + 3x - 2 = x^2 + 5x - 3
Q: Can I use the commutative property to find the difference of polynomials?
A: Yes, you can use the commutative property to find the difference of polynomials. The commutative property states that for any real numbers a and b:
a + b = b + a
You can use this property to rearrange the terms inside the polynomials.
For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2) = 3x^2 + 2x - 1 - 2x^2 + 3x - 2 = x^2 + 5x - 3
Q: Can I use the associative property to find the difference of polynomials?
A: Yes, you can use the associative property to find the difference of polynomials. The associative property states that for any real numbers a, b, and c:
(a + b) + c = a + (b + c)
You can use this property to rearrange the terms inside the polynomials.
For example, if we have two polynomials:
p(x) = 3x^2 + 2x - 1 q(x) = 2x^2 - 3x + 2
The difference of p(x) and q(x) is:
p(x) - q(x) = (3x^2 + 2x - 1) - (2x^2 - 3x + 2) = 3x^2 + 2x - 1 - 2x^2 + 3x - 2 = x^2 + 5x - 3
Conclusion
In conclusion, finding the difference of polynomials involves distributing the negative sign to the terms inside the second polynomial, combining like terms, and simplifying the expression. You can use various properties, such as the distributive property, commutative property, and associative property, to simplify the expression.