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Is [tex]\( x-3 \)[/tex] a factor of [tex]\( f(x) = x^3 - 9x^2 + 22x - 3 \)[/tex]?

A. Yes, the remainder is 0, so [tex]\( x-3 \)[/tex] is a factor.
B. Yes, the remainder is 9, so [tex]\( x-3 \)[/tex] is a factor.
C. No, the remainder is 0, so [tex]\( x-3 \)[/tex] is NOT a factor.
D. No, the remainder is 9, so [tex]\( x-3 \)[/tex] is NOT a factor.

Sagot :

GinnyAnswer
To determine if [tex]\(x - 3\)[/tex] is a factor of the polynomial [tex]\(f(x) = x^3 - 9x^2 + 22x - 3\)[/tex], we need to perform polynomial division and find the remainder when [tex]\(f(x)\)[/tex] is divided by [tex]\(x - 3\)[/tex].

By applying the polynomial division process, we extract the remainder after dividing [tex]\(f(x)\)[/tex] by [tex]\(x - 3\)[/tex]. If [tex]\(x - 3\)[/tex] is truly a factor of [tex]\(f(x)\)[/tex], the remainder must be zero.

Upon performing the polynomial division, we find that the remainder is 9.

Since the remainder of this division is not zero (specifically, it is 9), this indicates that [tex]\(x - 3\)[/tex] is not a factor of [tex]\(f(x)\)[/tex].

Thus, the correct answer is:
d. No, the remainder is 9 so [tex]\(x-3\)[/tex] is NOT a factor.
Hi1315

Answer:

No, the remainder is 9, so  x-3  is NOT a factor.

Step-by-step explanation:

To determine if  x-3  is a factor of  [tex]f(x) = x^3 - 9x^2 + 22x - 3[/tex] , we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial  f(x)  is divided by  x-a , the remainder of this division is  f(a) .

Let's evaluate  f(3) :

[tex]f(3) = 3^3 - 9 \cdot 3^2 + 22 \cdot 3 - 3[/tex]

f(3) = 27 - 81 + 66 - 3

f(3) = 27 - 81 + 66 - 3

f(3) = 9

Since the remainder is 9,  x-3  is not a factor of f(x).

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