Is (3x-2) a factor of 3x³+ x²-20x +12 ?
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Solution!!
The concept of factor and remainder theorem has to be used here.
(3x - 2) = 0
3x = 2
x = 2/3
f(x) = 3x³ + x² - 20x + 12
f(2/3) = 3(2/3)³ - (2/3)² - 20(2/3) + 12
0 = 3(8/27) - (4/9) - (40/3) + 12
0 = (8/9) - (4/9) - (40/3) + 12
0 = (8 - 4)/9 - (40/3) + 12
0 = (4/9) - (40/3) + 12
0 = (4 - 120 + 108)/9
0 ≠ -8/9
Hence, (3x - 2) is not a factor of the polynomial 3x³ - x² - 20x + 12.
Factor Theorem → When a polynomial f(x) is divided by x - a, the remainder = f(a). And, if remainder f(a) = 0; x - a is a factor of the polynomial f(x).
Remainder Theorem → If f(x), a polynomial in x, is divided by (x - a), the remainder = f(a) e.g. If f(x) is divided by (x - 3), the remainder is f(3).
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