Final answer:
The product of the polynomials 2x^2 + 3x − 6 and x^2 − 4x + 2 is 2x^4 − 5x^3 − 16x^2 + 30x − 12, obtained by polynomial multiplication and combining like terms.
Step-by-step explanation:
The product of two polynomials 2x^2 + 3x − 6 and x^2 − 4x + 2 is found by multiplying each term in the first polynomial by each term in the second polynomial. This process is also known as polynomial multiplication or the FOIL method (First, Outside, Inside, Last).
To perform this multiplication, we multiply each term in the first polynomial by each term in the second, which is:
- (2x^2) × (x^2) = 2x^4
- (2x^2) × (− 4x) = − 8x^3
- (2x^2) × (2) = 4x^2
- (3x) × (x^2) = 3x^3
- (3x) × (− 4x) = − 12x^2
- (3x) × (2) = 6x
- (− 6) × (x^2) = − 6x^2
- (− 6) × (− 4x) = 24x
- (− 6) × (2) = − 12
Then we combine like terms to get the final result:
2x^4 − 8x^3 + 3x^3 − 8x^2 − 12x^2 + 4x^2 + 24x + 6x − 12
2x^4 − 5x^3 − 16x^2 + 30x − 12