Question Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.
To demonstrate the Closure Property of multiplication of polynomials, we need to show that when we multiply any two polynomials together, the resulting product is also a polynomial.
Let's consider the polynomials:
P(x) = 2x^3 + 3x^2 - 5x + 4
Q(x) = 4x^2 - x + 2
If we multiply these polynomials together, we get:
P(x) * Q(x) = (2x^3 + 3x^2 - 5x + 4) * (4x^2 - x + 2)
Using the distributive property, we can multiply each term of the first polynomial by each term of the second polynomial:
P(x) * Q(x) = (2x^3 * 4x^2) + (2x^3 * -x) + (2x^3 * 2) + (3x^2 * 4x^2) + (3x^2 * -x) + (3x^2 * 2) + (-5x * 4x^2) + (-5x * -x) + (-5x * 2) + (4 * 4x^2) + (4 * -x) + (4 * 2)
Simplifying each term:
P(x) * Q(x) = 8x^5 - 2x^4 + 4x^3 + 12x^4 - 3x^3 + 6x^2 - 20x^3 + 5x^2 - 10x + 16x^2 - 4x + 8
Combining like terms:
P(x) * Q(x) = 8x^5 + 10x^4 - 9x^3 + 27x^2 - 14x + 8
We can see that the product of P(x) and Q(x) is also a polynomial, demonstrating the Closure Property of multiplication of polynomials.