Find rational roots of the polynomial f(x) = 2x³+x²-7x-6.
Answers
Given : f(x) = 2x³ + x² - 7x - 6.
We see that f(x) is a cubic polynomial (ax³ + bx² + cx + d ) with integer coefficients. If b/c is a rational root in lowest term, then the value of b have the factors of 6 which are ±1 , ±2 , ±3, ±6 and values of c have the factors of 2 which are ±1 , ±2.
Hence, the possible rational roots of f(x) are: ±1 , ±2 , ±3, ±6, ±1/2 , ±3/2
Now, On putting x = -1 in f(x) :
f (-1) = 2 (-1)³ + (-1)² - 7 (-1) - 6
= - 2 + 1 + 7 - 6
= - 1 + 1
= 0
On putting x = 2 in f(x) :
f (2) = 2 (2)³ + (2)² - 7 (2) - 6
= 2 × 8 + 4 - 14 - 6
= 16 + 4 - 14 - 6
= 20 - 20
= 0
On putting x = - 3/2 in f(x) :
f (- 3/2 ) = 2 (- 3/2)³ + (-3/2)² - 7 (- 3/2) - 6
= 2 × - 27/8 + 9/4 + 21/2 - 6
= - 27/4 + 9/4 + 21/2 - 6
= (-27 + 9)/4 + (21 - 12)/2
= - 18/4 + 9/2
= - 9/2 + 9/2
= 0
Hence, -1, 2, -3/2 are the rational roots of f (x).
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