Bot GPT 4-o mini To find the product of the polynomials \(2x - 3\) and \(x^2 + 6x - 2\), we will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions.
The polynomial multiplication can be computed as follows:
\[ (2x - 3)(x^2 + 6x - 2) \]
We can expand this:
-
Multiply \(2x\) by each term in the second polynomial:
- \(2x \cdot x^2 = 2x^3\)
- \(2x \cdot 6x = 12x^2\)
- \(2x \cdot (-2) = -4x\)
-
Multiply \(-3\) by each term in the second polynomial:
- \(-3 \cdot x^2 = -3x^2\)
- \(-3 \cdot 6x = -18x\)
- \(-3 \cdot (-2) = 6\)
Now we can combine all the results:
\[ 2x^3 + 12x^2 - 4x - 3x^2 - 18x + 6 \]
Next, we will combine like terms:
- For \(x^2\): \(12x^2 - 3x^2 = 9x^2\)
- For \(x\): \(-4x - 18x = -22x\)
Putting it all together, we have:
\[ 2x^3 + 9x^2 - 22x + 6 \]
Thus, the resulting polynomial is:
\[ 2x^3 + 9x^2 - 22x + 6 \]
Now we can match this result with one of the provided options:
C: \(2x^3 + 9x^2 - 22x + 6\)
Therefore, the correct answer is C.
1 answer