To find the product [tex]\((x - 3) \left(2x^2 - 5x + 1\right)\)[/tex], we will expand the expression step by step.
First, let's distribute [tex]\( (x - 3) \)[/tex] to each term in the polynomial [tex]\( 2x^2 - 5x + 1 \)[/tex]:
1. Distribute [tex]\( x \)[/tex] to each term in [tex]\( 2x^2 - 5x + 1 \)[/tex]:
[tex]\[
x \cdot 2x^2 + x \cdot (-5x) + x \cdot 1 = 2x^3 - 5x^2 + x
\][/tex]
2. Distribute [tex]\( -3 \)[/tex] to each term in [tex]\( 2x^2 - 5x + 1 \)[/tex]:
[tex]\[
-3 \cdot 2x^2 + (-3) \cdot (-5x) + (-3) \cdot 1 = -6x^2 + 15x - 3
\][/tex]
3. Combine the results from the two distributions:
[tex]\[
2x^3 - 5x^2 + x + (-6x^2 + 15x - 3)
\][/tex]
4. Combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
-5x^2 - 6x^2 = -11x^2
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
x + 15x = 16x
\][/tex]
Therefore, the expanded expression is:
[tex]\[
2x^3 - 11x^2 + 16x - 3
\][/tex]
So, the product of [tex]\( (x - 3) (2x^2 - 5x + 1) \)[/tex] is:
[tex]\[
2x^3 - 11x^2 + 16x - 3
\][/tex]
Hence, the correct answer is:
[tex]\[
2x^3 - 11x^2 + 16x - 3
\][/tex]
