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Find the product of [tex][tex]$(4x - 3)\left(2x^2 - 7x + 1\right)$[/tex][/tex].

A. [tex]$8x^3 - 22x^2 + 17x - 3$[/tex]
B. [tex]$8x^3 + 8x^2 + 4x - 3$[/tex]
C. [tex][tex]$8x^3 - 34x^2 + 25x - 3$[/tex][/tex]
D. [tex]$8x^3 - 42x^2 + 25x - 3$[/tex]


Sagot :

To find the product of the polynomials [tex]\((4x - 3)(2x^2 - 7x + 1)\)[/tex], we'll use the distributive property (also known as the FOIL method for binomials). Let's perform the multiplication step-by-step.

Given:
[tex]\[ (4x - 3)(2x^2 - 7x + 1) \][/tex]

First, distribute [tex]\(4x\)[/tex] to each term in the second polynomial:
[tex]\[ 4x \cdot 2x^2 = 8x^3 \][/tex]
[tex]\[ 4x \cdot (-7x) = -28x^2 \][/tex]
[tex]\[ 4x \cdot 1 = 4x \][/tex]

Next, distribute [tex]\(-3\)[/tex] to each term in the second polynomial:
[tex]\[ -3 \cdot 2x^2 = -6x^2 \][/tex]
[tex]\[ -3 \cdot (-7x) = 21x \][/tex]
[tex]\[ -3 \cdot 1 = -3 \][/tex]

Now, combine all these terms:
[tex]\[ 8x^3 - 28x^2 + 4x - 6x^2 + 21x - 3 \][/tex]

Group together the like terms:
[tex]\[ 8x^3 + (-28x^2 - 6x^2) + (4x + 21x) - 3 \][/tex]
[tex]\[ 8x^3 - 34x^2 + 25x - 3 \][/tex]

So, the product of [tex]\((4x - 3)(2x^2 - 7x + 1)\)[/tex] is:
[tex]\[ 8x^3 - 34x^2 + 25x - 3 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{8x^3 - 34x^2 + 25x - 3} \][/tex]
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