Use the following multiplication problem to show how multiplying polynomials is a closed system.
(x2−1)(−2x+5)
To show that multiplying polynomials is a closed system, we need to demonstrate that the result of multiplying two polynomials is also a polynomial.
Given the multiplication problem (x^2−1)(−2x+5), we can expand it using the distributive property:
(x^2)(-2x) + (x^2)(5) + (-1)(-2x) + (-1)(5)
Simplifying this expression:
-2x^3 + 5x^2 + 2x - 5
The result of multiplying the two polynomials is -2x^3 + 5x^2 + 2x - 5, which is a polynomial. Therefore, multiplying polynomials is a closed system.
Multiplying polynomials is like trying to combine two great comedians and seeing what kind of hilarity ensues. Let's take a look at the problem at hand. We have the polynomial (x^2 - 1) and the polynomial (-2x + 5). Now, let's combine these two funny characters and see what happens!
(x^2 - 1)(-2x + 5)
Now, we'll use the distributive property to bring these two comedy acts together and create a brand new show:
-2x * x^2 + 5 * x^2 - 2x * -1 + 5 * -1
So, we have -2x^3 + 5x^2 + 2x - 5. This is our final result, and guess what? It's still a polynomial! See how we started with polynomials and ended up with a polynomial again? That's what makes multiplying polynomials a closed system - it always stays within the realm of polynomials. It's like comedians staying in their lane, sticking to what they know and keeping the laughs going!
To show that multiplying polynomials is a closed system, we need to demonstrate that when we multiply two polynomials, the result is always another polynomial. Let's work through the given multiplication problem step-by-step to illustrate this:
Step 1: Multiply the first terms of each polynomial.
(x^2) * (-2x) = -2x^3
Step 2: Multiply the outer terms of each polynomial.
(x^2) * 5 = 5x^2
Step 3: Multiply the inner terms of each polynomial.
(-1) * (-2x) = 2x
Step 4: Multiply the last terms of each polynomial.
(-1) * 5 = -5
Step 5: Combine the results from steps 1 to 4.
-2x^3 + 5x^2 + 2x - 5
As we can see, the final result is a polynomial (-2x^3 + 5x^2 + 2x - 5). This demonstrates that multiplying the polynomials (x^2 - 1) and (-2x + 5) produces another polynomial. Hence, multiplying polynomials is a closed system.
To show that multiplying polynomials is a closed system, we need to demonstrate that multiplying two polynomials results in another polynomial.
To multiply two polynomials, such as (x^2 - 1) and (-2x + 5), we can use the distributive property. This involves multiplying each term in the first polynomial by each term in the second polynomial and adding the resulting terms together.
Let's break down the multiplication step by step:
First, multiply the first term of the first polynomial (x^2) with each term of the second polynomial (-2x + 5):
x^2 * -2x = -2x^3 (multiply the coefficients: 1 * -2 = -2 and add the exponents: x^2 * x = x^3)
x^2 * 5 = 5x^2 (multiply the coefficients: 1 * 5 = 5)
Next, multiply the second term of the first polynomial (-1) with each term of the second polynomial:
-1 * -2x = 2x (multiply the coefficients: -1 * -2 = 2)
-1 * 5 = -5 (multiply the coefficients: -1 * 5 = -5)
Now, we can combine the like terms by adding or subtracting:
-2x^3 + 5x^2 + 2x - 5
As we can see, the result of multiplying the two polynomials is another polynomial, -2x^3 + 5x^2 + 2x - 5. Therefore, multiplying polynomials is closed, meaning that the product of two polynomials is always a polynomial.