when a polynominal f x is divide by x^2 - 5 the quotient is x^3 - 2x -3 and reminder is 0 find the polynominal and all its zeros
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Answers
Correct Question :- when a polynominal f x is divide by x^2 - 5 the quotient is x^² - 2x -3 and reminder is 0.. find the polynominal and all its zeros ..
Solution :-
Dividend = Quotient x Divisor + Remainder .
Given :-
→ Divisor = x² - 5
→ Quotient = x² - 2x - 3
→ Remainder = 0
So,
→ Dividend = (x² - 5)(x² - 2x - 3) + 0
or,
→ f(x) = (x² - 5)(x² - 2x - 3)
→ f(x) = x⁴ - 2x³ - 3x² - 5x² + 10x + 15
→ f(x) = x⁴ - 2x³ - 8x² + 10x + 15 (Ans).
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Zeros Of Polynomial :-
→ (x² - 5)(x² - 2x - 3) = 0
→ {x² - (√5)²} [ x² -3x + x - 3 ] = 0
→ (x + √5)(x - √5) [ x(x - 3) + 1(x - 3) ] = 0
→ (x + √5)(x - √5) [ (x - 3)(x + 1) ] = 0
→ (x + √5)(x - √5)(x - 3)(x + 1) = 0
putting all Equal to Zero Now,
→ x + √5 = 0
→ x = -(√5)
→ x - √5 = 0
→ x = √5
→ x - 3 = 0
→ x = 3 .
→ x + 1 = 0
→ x = (-1) .
Hence, Zeros of Given Polynomial are [ ±√5 , 3 & (-1) ] .
Correct Question -
When a Polynomial f(x) is divided by x² - 5 the Quotient is x² - 2x -3 and reminder is 0. Find the polynomial and all it's zeros.
Given →
Divider = x² - 5
Quotient = x² - 2x - 3
Reminder = 0
To Find Out :-
The polynomial f(x) and it's zeros.
Solution :-
We know that -
- Dividend = Quotient × Divider + Reminder
So ,
→ Dividend f (x) = (x² - 2x - 3) × (x² - 5) + 0
→
The Zeros of This Polynomial :-
The polynomial is -
We can write it as -
→ (x² - 2x - 3) × (x² - 5)
→ (x²- 3x + x -3) (x² - 5)
→ [x(x - 3) + 1 (x - 3)] (x²-5)
→ (x + 1)(x - 3)(x² - 5)
First Zero :-
Second Zero -
x + 1 = 0
Third Zero -
x - 3 = 0
→ So the zeros of this polynomial are √5 , -1, 3.