In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:
f(x) = 2x³ -9x²+x+12, g(x) = 3-2x
Answers
Given : f(x) = 2x³ - 9x² + x + 12, g(x) = 3 - 2x
To find : Whether polynomial g(x) is a factor of polynomial f(x) or, not.
In order to find whether g (x) = 3 - 2x is a factor of polynomial f (x) or not, it is sufficient to prove that f (3/2) = 0 :
Now,
f(x) = 2x³ - 9x² + x + 12
f (3/2) = 2(3/2)³ - 9(3/2)² + (3/2) + 12
f (3/2) = 2 × 27/8 - 9 × 9/4 + 3/2 + 12
f (3/2) = 27/4 - 81/4 + 3/2 + 12
f (3/2) = (27 - 81 + 6 + 48)/4
f (3/2) = (- 54 + 54)/4
f (3/2) = 0/4
f (3/2) = 0
Hence, g (x) is a factor of f (x).
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g(x) = 3-2x
=>3-2x=0
=>2x=-3
=>x=-3/-2
=>x=3/2
Now,
f(x) = 2x³ - 9x² + x + 12
putting the value of x
f (3/2) = 2(3/2)³ - 9(3/2)² + (3/2) + 12
f (3/2) = 2 × 27/8 - 9 × 9/4 + 3/2 + 12
f (3/2) = 27/4 - 81/4 + 3/2 + 12
f (3/2) = (27 - 81 + 6 + 48)/4
f (3/2) = (- 54 + 54)/4
f (3/2) = 0/4
f (3/2) = 0
Learn more:
Show that (x-2), (x+3) and (x-4) are factors of x³-3x²-10x+24.
brainly.in/question/15904222