verify that 2,3,-5/3 is the zeroes of the cubic polynomial p(x)=3x^3-10x^2-7x+30 and then verify the relationship between the zeroes and the coefficients.
Answers
Answer:
Step-by-step explanation:
We have to verify that 2, 3 , -5/3 is the zeroes of the cubic polynomial p(x)= 3x³ - 10x² - 7x + 30 and have to verify the relationship between the zeroes and the coefficients.
solution : p(2) = 3(2)³ - 10(2)² - 7(2) + 30
= 3 × 8 - 10 × 4 - 14 + 30
= 24 - 40 - 14 + 30
= 0
so 2, is a zero of p(x)
p(3) = 3(3)³ - 10(3)² - 7(3) + 30.
= 3 × 27 - 10 × 9 - 21 + 30
= 81 - 90 - 21 + 30
= 0
so 3, is also a zero of p(x).
p(-5/3) = 3(-5/3)³ - 10(-5/3)² - 7(-5/3) + 30
= 3 × -125/27 - 10 × 25/9 + 35/3 + 30
= -125/9 - 250/9 + 35/3 + 30
= -(375)/9 + 35/3 + 30
= -125/3 + 35/3 + 30
= (-125 + 35)/3 + 30
= -30 + 30
= 0
so -5/3, is also a zero of p(x).
therefore 2, 3, -5/3 are the zeroes of polynomial p(x).
now sum of roots = -coefficient of x²/coefficient of x³
LHS = 2 + 3 - 5/3 = (15 - 5)/3 = 10/3
RHS = -(-10)/3 = 10/3
LHS = RHS
product of roots = -constant/coefficient of x³
LHS = (2)(3)(-5/3) = -10
RHS = -(30/3) = -10
LHS = RHS
sum of products of two consecutive roots = coefficient of x²/coefficient of x³
LHS = (2)(3) + (3)(-5/3) + (-5/3)(2)
= 6 - 5 -10/3
= (3 - 10)/3 = -7/3
RHS = (-7)/3 = -7/3
LHS = RHS
hence verified.