find the zeros of 2^x^3 + 5^x^2 - 9x - 18 if it is given that the product of its two zeros is -3
Answers
Step-by-step explanation:
-3. 2. 5 -9. -18
0. -6 3 18
2 -1 -6 | 0
-------
other factors is 2x^2-x-6=0
(x-2)(2x+3)
x=2,x=-3/2
Answer:
Let f( x)= 2x³+5x²-9x-18
Since f(x) is a cubic polynomial, then it must have three zeros.
Let a,b and y are the zeros of f(x).
Then, we get
a×b= (-3). .........I
Now for a cubic polynomial,we know that
Product of the zeros=
-Constant term/coefficient of x³
so, a× b×y= -(-18)/2
a×b×y=9
(-3)×y=9. from EQ.I
y= 9/(-3)
y= (-3)
And sum of the product of zeros taken two at a(-5) time= coefficient of x/coefficient of x³
so,
ab+by+ay= (-9)/2
(-3)+b(-3)+a(-3)= -9/2
-3a-3b= 3-(9/2)
-3(a+b)= -3/2
(a+b)= 1/2. .........2
now by squaring both sides ,we get
(a+b)²= (1/2)²
a²+b²+2ab= 1//4
(a-b)²+2ab+2ab= 1/4. [a²+b²=(a-b)²+2ab]
(a-b)²+4ab= 1/4
(a-b)²+4×(-3)= 1/4
(a-b)²-12 = 1/4
(a-b)²= 12+1/4
(a-b)²=49/4
(a-b)= √49/4
(a-b)= 7/2. ..............3
By adding EQ.s 2 and 3, we get as
(a+b)+(a-b)= 1/2+7/2
2a = 4
so, a=2
By putting a in EQ.i we get
a×b= (-3)
2×b=(-3)
so, b= (-3)/2
Hence ,the zeros of the given polynomial are 2,(-3/2) and (-3).
Hope you will enjoy the solution bestly.