What is the remainder when 3t^3-2t+2 is divided by 3t+1?

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answers:
Ian H say: Dividing a polynomial f(x) by x – a, (with degree 1), the result takes the form
f(x) = (x – a)g(x) + c, (where remainder c has degree 0, i.e. it is a constant).
Now if we let x = a, we get a consequence known as the Remainder Theorem
f(a) = c
We find the remainder of dividing f(x) by x – a, by substituting x = a into f(x)
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What is the remainder when 3t^3 - 2t + 2 is divided by 3t + 1?
When 3t + 1 = 0, t = -1/3, so that is what we substitute;
f(-1/3) = 3(-1/3)^3 - 2(-1/3) + 2 = = -1/9 + 2/3 + 2 = 23/9

Extra note: f(x) = (3t^3 - 2t - 5/9) + 23/9
and dividing by 3t + 1 gives t^2 – t/3 with remainder 23/9
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sepia say: (3t^3 - 2t + 2) / (3t + 1)
Quotient and remainder:
3 t^3 - 2 t + 2 = (t^2 - t/3 - 5/9) × (3 t + 1) + 23/9
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khalil say: 3t+1 = 0

t = -1/3

r = f(-1/3) = 3(-1/3)³ -2(-1/3) +2

r = 23/9
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la console say: (3t³ - 2t + 2) / (3t + 1)

First term: 3t³/3t = t²
t².(3t + 1) = 3t³ + t²
Rest:
= (3t³ - 2t + 2) - (3t³ + t²)
= 3t³ - 2t + 2 - 3t³ - t²
= - t² - 2t + 2

Second term: - t²/3t = - (1/3).t
- (1/3).t.(3t + 1) = - t² - (1/3).t
Rest:
= (- t² - 2t + 2) - [- t² - (1/3).t]
= - t² - 2t + 2 + t² + (1/3).t
= - (5/3).t + 2

Third term: - (5/3).t/3t = - 5/9
- (5/9).(3t + 1) = - (5/3).t - (5/9)
Rest:
= [- (5/3).t + 2] - [- (5/3).t - (5/9)]
= - (5/3).t + 2 + (5/3).t + (5/9)
= 2 + (5/9)
= 23/9 ← this is the remainder

3t³ - 2t + 2 = [(3t + 1).[t² - (1/3).t - (5/9)] + (23/9)
(3t³ - 2t + 2) / (3t + 1) = t² - (1/3).t - (5/9) + { 23/[9.(3t + 1)] }
(3t³ - 2t + 2) / (3t + 1) = t² - (1/3).t - (5/9) + [23/(27t + 9)]
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nbsale say: You get a quotient of t^2 -t/3 -5/9 and a remainder of 23/9
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llaffer say: If dividing by (3t + 1), the remainder is the same as if t = -1/3 (what makes that expression a zero).

So solve for f(1/3) and that's your remainder:

f(t) = 3t³ - 2t + 2
f(-1/3) = 3(-1/3)³ - 2(-1/3) + 2
f(-1/3) = 3(-1/27) - 2(-1/3) + 2
f(-1/3) = -1/9 + 2/3 + 2
f(-1/3) = -1/9 + 6/9 + 18/9
f(-1/3) = 23/9

I get the same as Morningfox.
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tiyt say: try an online cal
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Morningfox say: I get a remainder of 23/9.
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