verify that the numbers given along side The Cubic polynomials below are thier zeros also verify the relationship between the zeros and the coefficient in each case
2x³+x²-5x+2:(½,1,-2)
Answers
Step-by-step explanation:
Given,
p(x) : 2x³ + x² - 5x + 2
If 1/2 is a zero of p(x), Then, p(1/2) should be zero.
p(1/2) = 2×(1/2)³ + (1/2)² -5×(1/2) + 2
= 2×(1/8) + 1/4 - 5/2 + 2
= 1/4 +1/4 - 5/2 + 2
= 2/4 -5/2 + 2
= 1/2 - 5/2 + 2
= -4/2 + 2
= -2 + 2
= 0
Hence, 1/2 is a zero of given polynomial
Similarly,
p(1) = 2×(1)³ + 1² -5×1 + 2
=2 + 1 - 5 + 2
= 3 - 5 + 2
= 2
And,
p(-2) = 2×(-2)³ + (-2)² -5(-2) + 2
= -2×8 + 4 + 10 + 2
= -16 + 14 + 2
= -2 + 2
= 0
Hence, 1 and -2 are also zeroes of given polynomial
Now, We have to verify relations between zeroes:-
Comparing given polynomial with standard form of equation of 3rd degree,
ax³ + bx + c + d
a = 2
b = 1
c = -5
d = 2
Sum of zeroes = -b/a
1/2 + 1 + (-2) = -1/2
1/2 - 1 = -1/2
-1/2 = -1/2
L.H.S. = R.H.S.
Product of zeroes = d/a
1/2 × 1 × (2) = 2/2
1 = 1
L.H.S. = R.H.S.