without actual division, prove that (2x^4-6x^3+3x^2+3x-2) is exactly divisible by (x^2-3x+2)
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Let f(x) = x4 + 2x3 -2x2 + 2x - 3
g(x) = x2 + 2x - 3
= x(x + 3) - 1(x + 3) = (x - 1) (x + 3)
Now f(x) will be exactly divisible by g(x) if it is exactly divisible by (x - 1) as well as (x + 3)
i.e. if f(1) = 0 and f ( -3) = 0
Now f(1) = 14 + 2.13 -2.12 + 2.1 - 3
= 1 + 2 - 2 + 2 - 3 = 0
=> (x - 1) is a factor of f(x)
f ( -3) = (-3)4 + 2.(-3)3 -2.(-3)2 + 2.(-3) - 3
= 81 - 54 - 18 - 6 - 3 = 0
=> (x + 3) is a factor of f(x).
=> (x - 1) (x + 3) divides f (x) exactly
Therefore, x2 + 2x - 3 is a factor of f(x)
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Given f(x) = 2x^4 - 6x^3 + 3x^2 + 3x - 2
x^2 - 3x + 2 = x^2 - x - 2x + 2
= x(x - 1) -2(x - 1)
= (x - 1)(x - 2)
If (x - 1) and (x - 2) are the factors of f(x).Then f(x) is divisible by x^2 - 3x + 2.
if f(1) = 0 and f(2) = 0, then f(x) is exactly divisible by x^2 - 3x + 2.
f(1) = 2(1)^4 - 6(1)^3 + 3(1)^2 + 3(1) - 2
= 2 - 6 + 3 + 3 - 2
= 0. -------- (1)
f(2) = 2(2)^4 - 6(2)^3 + 3(2)^2 + 3(2) - 2
= 2(16) - 6(8) + 3(4) + 6 - 2
= 32 - 48 + 12 + 6 - 2
= 0. ------- (2)
From (1) & (2), we get
f(1) = 0 and f(2) = 0.
Therefore f(x) is exactly divisible by x^2 - 3x + 2.
Read more on Brainly.in - https://brainly.in/question/2287012#readmore
g(x) = x2 + 2x - 3
= x(x + 3) - 1(x + 3) = (x - 1) (x + 3)
Now f(x) will be exactly divisible by g(x) if it is exactly divisible by (x - 1) as well as (x + 3)
i.e. if f(1) = 0 and f ( -3) = 0
Now f(1) = 14 + 2.13 -2.12 + 2.1 - 3
= 1 + 2 - 2 + 2 - 3 = 0
=> (x - 1) is a factor of f(x)
f ( -3) = (-3)4 + 2.(-3)3 -2.(-3)2 + 2.(-3) - 3
= 81 - 54 - 18 - 6 - 3 = 0
=> (x + 3) is a factor of f(x).
=> (x - 1) (x + 3) divides f (x) exactly
Therefore, x2 + 2x - 3 is a factor of f(x)
4.0
3 votes
THANKS
4
CommentsReport
THE BRAINLIEST ANSWER!
siddhartharao77Genius
This is a Verified Answer
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Verified Answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but Verified Answers are the finest of the finest.
Given f(x) = 2x^4 - 6x^3 + 3x^2 + 3x - 2
x^2 - 3x + 2 = x^2 - x - 2x + 2
= x(x - 1) -2(x - 1)
= (x - 1)(x - 2)
If (x - 1) and (x - 2) are the factors of f(x).Then f(x) is divisible by x^2 - 3x + 2.
if f(1) = 0 and f(2) = 0, then f(x) is exactly divisible by x^2 - 3x + 2.
f(1) = 2(1)^4 - 6(1)^3 + 3(1)^2 + 3(1) - 2
= 2 - 6 + 3 + 3 - 2
= 0. -------- (1)
f(2) = 2(2)^4 - 6(2)^3 + 3(2)^2 + 3(2) - 2
= 2(16) - 6(8) + 3(4) + 6 - 2
= 32 - 48 + 12 + 6 - 2
= 0. ------- (2)
From (1) & (2), we get
f(1) = 0 and f(2) = 0.
Therefore f(x) is exactly divisible by x^2 - 3x + 2.
Read more on Brainly.in - https://brainly.in/question/2287012#readmore
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