Question

Question Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.

Answer 1

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.