Question

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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Answer 1

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) ] .

Answer 2

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.

$\rule{200}{1}$

Solution :-

We know that -

• Dividend = Quotient × Divider + Reminder

So ,

→ Dividend f (x) = (x² - 2x - 3) × (x² - 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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