Please answer question 17
Answers
Answer:
The third factor is x+1.
The complete factorization is ( x - 2 ) ( 3x + 1 ) ( x + 1 )
Step-by-step explanation:
Let f(x) = 3x³ - 2x² - 7x - 2.
f(2) = 3×2³ - 2×2² - 7×2 - 2 = 3×8 - 2×4 - 14 - 2 = 24 - 8 - 14 - 2 = 0.
Therefore x-2 is a factor of f(x).
f(-1/3) = 3×(-1/3)³ - 2×(-1/3)² - 7×(-1/3) - 2
= -1/9 - 2/9 + 7/3 - 2
= -3/9 + 7/3 - 2
= -1/3 + 7/3 - 2
= 6/3 - 2
= 2 - 2
= 0
Therefore (x - -1/3) = x+1/3 is a factor of f(x). Multiplying by 3 to get rid of the fractional part, 3(x+1/3) = 3x+1 is a factor of f(x).
As f(x) has degree 3, there is one more linear factor. So:
f(x) = 3x³ - 2x² - 7x - 2 = ( x - 2 ) ( 3x + 1 ) ( ax + b )
for some a and b.
Equating the coefficients of x³ gives
3 = 3a => a = 1
Equating the constant coefficient gives
-2 = -2b => b = 1
Therefore the third factor is x+1.
The complete factorization is then:
f(x) = ( x - 2 ) ( 3x + 1 ) ( x + 1 )
hope my answer will be right
