Math, asked by kavithakjadhav107266, 1 year ago

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

Answered by RvChaudharY50
28

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).

_________________

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) ] .

Answered by Anonymous
96

Correct Question -

When a Polynomial f(x) is divided by - 5 the Quotient is x² - 2x -3 and reminder is 0. Find the polynomial and all it's zeros.

Given

Divider = - 5

Quotient = x² - 2x - 3

Reminder = 0

To Find Out :-

The polynomial f(x) and it's zeros.

\rule{200}{1}

Solution :-

We know that -

  • Dividend = Quotient × Divider + Reminder

So ,

Dividend f (x) = (x² - 2x - 3) × ( - 5) + 0

f(x) =  {x}^{4}  - 2 {x}^{3}  - 3 {x}^{2}  - 5 {x}^{2}  + 10x + 15 \\ \boxed{\red {f(x) =  {x}^{4}  - 2 {x}^{3}  - 8 {x}^{2}  + 10x + 15}}

\rule{200}{1}

The Zeros of This Polynomial :-

The polynomial is -

f(x) =  {x}^{4}  - 2 {x}^{3}  - 8 {x}^{2}  + 10x + 15 \:

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)

\rule{200}{1}

First Zero :-

 {x}^{2}  - 5 = 0 \\  {x}^{2}  = 5 \\ \boxed{\green{ x =  \sqrt{5} }}

\rule{200}{1}

Second Zero -

x + 1 = 0

\boxed{\blue{x = -1 }}

\rule{200}{1}

Third Zero -

x - 3 = 0

\boxed{\purple{x = 3 }}

\rule{200}{1}

So the zeros of this polynomial are 5 , -1, 3.

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