What is the process of multiplying trinomials?

Multiplying trinomials can seem like a daunting task. The complexity can be reduced by breaking the problem down and multiplying each term separately. In this example, the trinomials x^3 + 4x^2 + 3x + 7 and 8x^3 + 6x^2 + 5x + 2 are multiplied.

1. Multiply the first term of the first trinomial with every term in the second trinomial

   Take x^3 and multiply it by every term in the second trinomial, 8x^3 + 6x^2 + 5x + 2. The formula is x^3(8x^3) + x^3(6x^2) + x^3(5x) + x^3(2). The answer is 8x^6 + 6x^5 + 5x^4 + 2x^3 and is used in a later step.

2. Multiply the second term of the first trinomial with every term in the second trinomial

   Take 4x^2 and multiply it by every term in the second trinomial, 8x^3 + 6x^2 + 5x + 2. The formula is 4x^2(8x^3) + 4x^2(6x^2) + 4x^2(5x) + 4x^2(2). The answer is 16x^5 + 8x^4 + 20x^3 + 8x^2 and is used in a later step.

3. Multiply the third term of the first trinomial with every term in the second trinomial

   Take 3x and multiply it by every term in the second trinomial, 8x^3 + 6x^2 + 5x + 2. The formula is 3x(8x^3) + 3x(6x^2) + 3x(5x) + 3x(2). The answer is 24x^4 + 18x^3 + 15 x^2 + 6x and is used in a later step.

4. Multiply the fourth term of the first trinomial with every term in the second trinomial

   Take 7 and multiply it by every term in the second trinomial, 8x^3 + 6x^2 + 5x + 2. The formula is 7(8x^3) + 7(6x^2) + 7(5x) + 7(2), which reduces to 56x^3 + 42x^2 + 35x + 14. This answer, along with the answers from the previous steps, is used in the next step.

5. Add the answers from the previous steps together

   Take the answers from the previous steps, 8x^6 + 6x^5 + 5x^4 + 2x^3, 16x^5 + 8x^4 + 20x^3 + 8x^2, 24x^4 + 18x^3 + 15x^2 + 6x and 56x^3 + 42x^2 + 35x + 14, and add the like terms together. Reduce the equation to 8x^6 + (6x^5 + 16x^5) + (5x^4 + 8x^4 + 24x^4) + (2x^3 + 20x^3 + 18x^3 + 56x^3) + (8x^2 + 15x^2 + 42x^2) + (6x + 35x) + 14 to get the final answer of 8x^6 + 22x^5 + 37x^4 + 96x^3 + 65x^2 + 41x + 14.