(x^5+2x^4+3x^2+x-3) (x^2+1)

Jasonbradley

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Jun 17, 2024

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Expanding the Polynomial: (x^5 + 2x^4 + 3x^2 + x - 3)(x^2 + 1)

This expression involves multiplying two polynomials: a fifth-degree polynomial (x^5 + 2x^4 + 3x^2 + x - 3) and a second-degree polynomial (x^2 + 1). To expand this expression, we'll use the distributive property (also known as FOIL for binomials).

Step-by-Step Expansion

1. Distribute the first term of the first polynomial (x^5) over the second polynomial:

   • x^5 * (x^2 + 1) = x^7 + x^5

2. Distribute the second term of the first polynomial (2x^4) over the second polynomial:

   • 2x^4 * (x^2 + 1) = 2x^6 + 2x^4

3. Distribute the third term of the first polynomial (3x^2) over the second polynomial:

   • 3x^2 * (x^2 + 1) = 3x^4 + 3x^2

4. Distribute the fourth term of the first polynomial (x) over the second polynomial:

   • x * (x^2 + 1) = x^3 + x

5. Distribute the fifth term of the first polynomial (-3) over the second polynomial:

   • -3 * (x^2 + 1) = -3x^2 - 3

6. Finally, combine all the terms:

   • x^7 + x^5 + 2x^6 + 2x^4 + 3x^4 + 3x^2 + x^3 + x - 3x^2 - 3

7. Simplify by combining like terms:

   • x^7 + 2x^6 + x^5 + 5x^4 + x^3 + x - 3

Result

Therefore, the expanded form of (x^5 + 2x^4 + 3x^2 + x - 3)(x^2 + 1) is x^7 + 2x^6 + x^5 + 5x^4 + x^3 + x - 3.