Show how multiplying polynomials is closed system (x^2-1)(-2x+5)
To show that multiplying polynomials is a closed system, we need to demonstrate that when we multiply two polynomials, the result is also a polynomial.
Let's multiply the given polynomials:
(x^2 - 1)(-2x + 5)
First, distribute the terms:
-2x(x^2 - 1) + 5(x^2 - 1)
Now, multiply each term:
-2x * x^2 + (-2x) * (-1) + 5 * x^2 + 5 * (-1)
Simplify:
-2x^3 + 2x + 5x^2 - 5
Now, combine like terms:
-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, we have shown that multiplying polynomials is a closed system.
To show that multiplying polynomials is a closed system, we need to demonstrate that when we multiply two polynomials, the result is also a polynomial.
Given the polynomials (x^2-1) and (-2x+5), we can find their product by performing the multiplication:
(x^2-1)(-2x+5)
To do this, we can use the FOIL method, which stands for First, Outer, Inner, Last. It helps us remember the steps to multiply two binomials.
First, we multiply the first terms of each binomial:
(x^2) * (-2x) = -2x^3
Next, we multiply the outer terms:
(x^2) * (5) = 5x^2
Then, we multiply the inner terms:
(-1) * (-2x) = 2x
Finally, we multiply the last terms:
(-1) * (5) = -5
Now, we collect the like terms and write out the equation:
-2x^3 + 5x^2 + 2x - 5
This expression is a polynomial because it is a sum or difference of monomials (terms with variables raised to non-negative integer exponents). Hence, we have shown that multiplying the polynomials (x^2-1) and (-2x+5) results in a polynomial. Therefore, multiplying polynomials is a closed system.
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 start with the given polynomials: (x^2-1) and (-2x+5). To find their product, we can use the distributive property.
We'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms, if any.
Multiply the first term of the first polynomial, x^2, by both terms of the second polynomial:
(x^2) * (-2x) = -2x^3
(x^2) * (5) = 5x^2
Next, multiply the second term of the first polynomial, -1, by both terms of the second polynomial:
(-1) * (-2x) = 2x
(-1) * (5) = -5
Now, we can combine the like terms obtained from the above multiplications:
-2x^3 + 5x^2 + 2x - 5
This resulting expression is the product of the two given polynomials. As we can see, it is a polynomial itself. Hence, multiplying polynomials is a closed system.
Therefore, the product of (x^2-1) and (-2x+5) is -2x^3 + 5x^2 + 2x - 5.