To find what must be added to the polynomial [tex]\(x^3 - 6x^2 + 11x - 8\)[/tex] so that [tex]\((x - 3)\)[/tex] becomes a factor of the polynomial, we need to determine the remainder when the given polynomial is divided by [tex]\((x - 3)\)[/tex].
According to the remainder theorem, if we evaluate the polynomial at [tex]\(x = 3\)[/tex], the result will be the remainder when the polynomial is divided by [tex]\((x - 3)\)[/tex].
Given the polynomial [tex]\(P(x) = x^3 - 6x^2 + 11x - 8\)[/tex], we compute [tex]\(P(3)\)[/tex]:
[tex]\[
P(3) = 3^3 - 6 \cdot 3^2 + 11 \cdot 3 - 8
\][/tex]
First, we calculate the individual terms:
[tex]\[
3^3 = 27
\][/tex]
[tex]\[
6 \cdot 3^2 = 6 \cdot 9 = 54
\][/tex]
[tex]\[
11 \cdot 3 = 33
\][/tex]
Next, we combine these results:
[tex]\[
P(3) = 27 - 54 + 33 - 8
\][/tex]
Now we simplify:
[tex]\[
27 - 54 = -27
\][/tex]
[tex]\[
-27 + 33 = 6
\][/tex]
[tex]\[
6 - 8 = -2
\][/tex]
The remainder when [tex]\(x^3 - 6x^2 + 11x - 8\)[/tex] is divided by [tex]\((x - 3)\)[/tex] is [tex]\(-2\)[/tex].
To make [tex]\((x - 3)\)[/tex] a factor of the polynomial, we need the remainder to be [tex]\(0\)[/tex]. Therefore, we must add [tex]\(2\)[/tex] to the polynomial [tex]\(x^3 - 6x^2 + 11x - 8\)[/tex].
Hence, the term that must be added is [tex]\(2\)[/tex]. So the updated polynomial will be:
[tex]\[
x^3 - 6x^2 + 11x - 8 + 2 = x^3 - 6x^2 + 11x - 6
\][/tex]
Therefore, [tex]\(2\)[/tex] must be added to [tex]\(x^3 - 6x^2 + 11x - 8\)[/tex] to make [tex]\((x - 3)\)[/tex] a factor of the polynomial.
