Final answer:
To simplify the expression (3x² - 7x - 2) (2c² - x - 1) to a polynomial in standard form, we need to distribute each term in the first binomial to each term in the second binomial and combine like terms. The simplified polynomial expression is -3x³ + 6c²x² + 4x² - 7c²x + 9x - 4c² + 2.
Step-by-step explanation:
To simplify the expression (3x² - 7x - 2) (2c² - x - 1) to a polynomial in standard form, we need to multiply the two binomials using the distributive property. Start by distributing the first term of the first binomial to each term in the second binomial:
3x² * 2c² = 6c²x²
3x² * -x = -3x³
3x² * -1 = -3x²
Now, distribute the second term of the first binomial to each term in the second binomial:
-7x * 2c² = -14c²x
-7x * -x = 7x²
-7x * -1 = 7x
Finally, distribute the third term of the first binomial to each term in the second binomial:
-2 * 2c² = -4c²
-2 * -x = 2x
-2 * -1 = 2
Now, combine like terms:
6c²x² - 3x³ - 3x² - 14c²x + 7x² + 7x - 4c² + 2x + 2
Putting the terms in descending order of degree, we get:
-3x³ + 6c²x² + 7x² - 3x² - 14c²x + 7x + 2x - 4c² + 2
This simplifies to:
-3x³ + 6c²x² + 4x² - 7c²x + 9x - 4c² + 2