use the remainder theorem to find the remainder when f(x) is divided by x - 3. Then use the factor theorem to determine whether x-3 is a factor of f(x) f(x)=2x^3-10x^2+11x+12
Given f(x)=2x^3-10x^2+11x+12
f(3)=2(3)^3-10(3)^2+11(3)+12 = 9
because f(3)not equal to 0
Therefore x-3 is not a factor of f(x)
Well, well, well, let's dive into some math humor, shall we?
To find the remainder when f(x) is divided by x - 3 using the remainder theorem, we simply need to substitute x = 3 into f(x) and see what hilarity unfolds. Prepare yourself!
f(x) = 2x^3 - 10x^2 + 11x + 12
Substituting x = 3:
f(3) = 2(3)^3 - 10(3)^2 + 11(3) + 12
= 2(27) - 10(9) + 33 + 12
= 54 - 90 + 33 + 12
= -3
So, the remainder when f(x) is divided by x - 3 is -3. Now, let's move on to the factor theorem, which has a comedy twist!
To determine whether x - 3 is a factor of f(x), we need to check if f(3) equals zero. Remembering the value we got earlier (-3), let's see if this equation delivers a punchline:
f(3) = -3
Whoops! It seems that f(3) is not equal to 0. Therefore, we can conclude that x - 3 is NOT a factor of f(x). It seems this polynomial has its own unique sense of mathematical humor!
To find the remainder when dividing f(x) by x - 3 using the remainder theorem, we substitute x - 3 into f(x) and calculate the result:
f(x) = 2x^3 - 10x^2 + 11x + 12
Substituting x - 3 into f(x):
f(x - 3) = 2(x - 3)^3 - 10(x - 3)^2 + 11(x - 3) + 12
Expanding and simplifying:
f(x - 3) = 2(x^3 - 9x^2 + 27x - 27) - 10(x^2 - 6x + 9) + 11(x - 3) + 12
= 2x^3 - 18x^2 + 54x - 54 - 10x^2 + 60x - 90 + 11x - 33 + 12
= 2x^3 - 28x^2 + 125x - 165
The remainder when f(x) is divided by x - 3 is 125x - 165.
To determine if x - 3 is a factor of f(x) using the factor theorem, we need to check if the remainder (125x - 165) is equal to zero when x = 3:
125(3) - 165 = 375 - 165 = 210
Since the remainder is not equal to zero when x = 3, we conclude that x - 3 is not a factor of f(x).
To find the remainder when dividing f(x) by (x - 3), we can use the remainder theorem. The remainder theorem states that if you divide a polynomial f(x) by (x - a), the remainder will be equal to f(a).
So, to find the remainder when f(x) is divided by (x - 3), we need to evaluate f(3).
First, let's find f(3) by substituting x = 3 into the expression f(x):
f(3) = 2(3)^3 - 10(3)^2 + 11(3) + 12
= 2(27) - 10(9) + 33 + 12
= 54 - 90 + 33 + 12
= 9
Therefore, the remainder when f(x) is divided by (x - 3) is 9.
Now, let's use the factor theorem to determine whether (x - 3) is a factor of f(x).
The factor theorem states that if f(a) = 0, then (x - a) is a factor of f(x).
Using the remainder theorem, we found that when f(x) is divided by (x - 3), the remainder is 9. Since the remainder is not zero, f(3) ≠ 0, which means (x - 3) is not a factor of f(x).
Therefore, (x - 3) is not a factor of f(x).