Use the following multiplication problem to show multiplying polynomials is a closed system
(X^2-1)(-2x+5)
To show that multiplying polynomials is a closed system, we need to demonstrate that the product of two polynomials is also a polynomial.
Let's multiply the given polynomials (X^2-1) and (-2x+5):
(X^2-1)(-2x+5)
= X^2(-2x) + X^2(5) + (-1)(-2x) + (-1)(5)
= -2x^3 + 5x^2 + 2x - 5
As we can see, the product is a polynomial (-2x^3 + 5x^2 + 2x - 5). 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 resulting expression is also a polynomial.
Let's begin by multiplying the given polynomials (X^2-1) and (-2x+5):
(X^2 - 1) * (-2x + 5)
To perform this multiplication, we will use the distributive property of multiplication over addition. We will first multiply each term of the first polynomial by the second polynomial and then combine like terms.
= (X^2 * -2x) + (X^2 * 5) + (-1 * -2x) + (-1 * 5)
Simplify each term:
= -2x^3 + 5X^2 + 2x - 5
The resulting expression, -2x^3 + 5X^2 + 2x - 5, is a polynomial since it is composed of powers of X (in this case, X^3, X^2, and X) multiplied by constants (-2, 5, and -5), and added or subtracted.
Therefore, we have shown that multiplying the polynomials (X^2 - 1) and (-2x + 5) resulted in a polynomial expression (-2x^3 + 5X^2 + 2x - 5). This demonstrates 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.
Let's multiply the given polynomials step-by-step:
Step 1: Distribute -2x to each term in the first polynomial and distribute 5 to each term in the same polynomial:
(-2x)(X^2) + (-2x)(-1) + (5)(X^2) + (5)(-1)
Step 2: Simplify each term by multiplying the coefficients and combining like terms:
-2x^3 + 2x + 5x^2 - 5
The result (-2x^3 + 2x + 5x^2 - 5) is a polynomial since it is a sum of terms containing variables raised to non-negative integer powers (x^3, x, x^2), multiplied by constants (-2, 2, 5), and combined using addition and subtraction. Therefore, the multiplication of the given polynomials is a closed system.