if x^2-9X+1=0 Find the value of x+1/x and x^3+1/x^3
Answers
Answer:
Step-by-step explanation:
The value of ( x + 1/x ) is 9.
The value of ( x³ + 1/x³ ) is 702.
Given: The polynomial: x² - 9x + 1 = 0
To Find: The values of ( x + 1/x ) and ( x³ + 1/x³ ).
Solution:
We shall make use of a formula to solve this numerical. The formula states that;
( a + b )³ = a³ + b³ + 3 × a × b × ( a + b ) .....(1)
Where a and b are integer values.
Coming to the numerical, we are given;
The polynomial: x² - 9x + 1 = 0
We can write it as;
x² + 1 = 9x
Dividing both sides by x, we get;
⇒ x + 1/x = 9
So, the value of x + 1/x is 9.
Now, we need to find the value of x³ + 1/x³. So, we put the respective values in (1) to get the required value;
( x + 1/x )³ = x³ + ( 1/x )³ + 3 × ( x ) × ( 1/x ) × ( x + 1/x )
⇒ ( x + 1/x )³ = x³ + ( 1/x )³ + 3 × ( x + 1/x )
⇒ ( 9 )³ = x³ + ( 1/x )³ + 3 × ( 9 )
⇒ 729 = x³ + ( 1/x )³ + 27
⇒ x³ + ( 1/x )³ = 729 - 27
⇒ x³ + ( 1/x )³ = 702
So, the value of x³ + 1/x³ is 702.
Hence,
The value of ( x + 1/x ) is 9.
The value of ( x³ + 1/x³ ) is 702.
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