Sure! Let's solve this step-by-step to find the other factor of the polynomial [tex]\(3x^2 - 10x + 8\)[/tex] given that one factor is [tex]\(3x - 4\)[/tex].
The polynomial can be represented as:
[tex]\[
3x^2 - 10x + 8
\][/tex]
Given factor:
[tex]\[
3x - 4
\][/tex]
Our task is to determine the other factor. Let's denote the other factor as [tex]\((x - a)\)[/tex].
Since [tex]\(3x - 4\)[/tex] is a factor of the polynomial, the product of [tex]\(3x - 4\)[/tex] and the other factor [tex]\((x - a)\)[/tex] should give us the original polynomial:
[tex]\[
(3x - 4)(x - a) = 3x^2 - 10x + 8
\][/tex]
To find [tex]\(a\)[/tex] (the value of the other factor), we can perform polynomial division or use factorization. However, we can simplify our approach by knowing that the original polynomial equals the product of its factors.
So if we have:
[tex]\[
(3x - 4)(x - a) = 3x^2 - 10x + 8
\][/tex]
We can find [tex]\(a\)[/tex] by simply expanding the left-side expression and matching it to our original polynomial.
First, expand the left-hand side:
[tex]\[
(3x - 4)(x - a) = 3x(x - a) - 4(x - a)
\][/tex]
[tex]\[
= 3x^2 - 3ax - 4x + 4a
\][/tex]
Combine like terms:
[tex]\[
= 3x^2 - (3a + 4)x + 4a
\][/tex]
This expanded polynomial must equal the original polynomial:
[tex]\[
3x^2 - 10x + 8
\][/tex]
Now, we compare the coefficients of corresponding terms on both sides:
[tex]\[
3x^2 - (3a + 4)x + 4a = 3x^2 - 10x + 8
\][/tex]
From the comparison of the coefficients, we can write the following system of equations:
[tex]\[
3a + 4 = 10
\][/tex]
[tex]\[
4a = 8
\][/tex]
Solving the first equation:
[tex]\[
3a + 4 = 10
\][/tex]
[tex]\[
3a = 6
\][/tex]
[tex]\[
a = 2
\][/tex]
Solving the second equation (which should also be true for our solution to be consistent):
[tex]\[
4a = 8
\][/tex]
[tex]\[
a = 2
\][/tex]
Thus, the other factor is:
[tex]\[
x - a = x - 2
\][/tex]
So the correct answer is:
[tex]\[
\boxed{x - 2}
\][/tex]
