Question

When a polynomial f(x) is divided by x2 – 5 the quotient is x2 – 2x – 3 and remainder is zero. Find the polynomial and all its zeroes

Answer 1

Given that,

g(x) = x^2 - 5

q(x) = x^2 - 2x - 3

r(x) = 0

To find,

f(x) = ?

Let x be the dividend or polynomial.

Here, everything is given except the polynomial. It's superb easy to find it. Just by using Euclid's division lemma which gives us a formula as,

$\boxed{\bold{\mathsf{Dividend = Divisor \times Quotient + Remainder}}}$

We been learning it since our lower classes but we don't know actually what is it? This's nothing but Euclid's division lemma.

So, by using Euclid's division lemma,

Dividend = Divisor $\times$ Quotient + Remainder

x = x^2 - 5 $\times$ x^2 - 2x - 3 + 0

x = (x^2 - 5)(x^2 - 2x - 3)

x = x^4 - 2x^3 - 3x - 5x^2 + 10x + 15

x = x^4 - 2x^3 - 5x^2 + (10x - 3x) + 15

x = x^4 - 2x^3 - 5x^2 + 7x + 15

But x is the polynomial or f(x).

$\boxed{\bold{\mathsf{\therefore f(x) = x^4 - 2x^3 - 5x^2 + 7x + 15}}}$

Answer 2

Answer:

Step-by-step explanation: