Question

Divide the polynomial P(x) by the polynomial g(x) and find the quotient and remainder. polynomial P(x) by the polynomial g(x) and find the quotient and remainder.p(x)=x⁴+x³+x²-2x-3(x)=-x²-2​

Answer 1

Answer:

Quotient: $x^2 + x + 3$

Remainder: 3

Step-by-step explanation:

We can use factor theorem

let g(y) = 0

$y^2 - 2 = 0\\y = \sqrt{2}$

P(y) = $4 + 2\sqrt{2} + 2 - 2\sqrt{2} - 3 = 3$

Therefore remainder = 3

Euclid's division lemma:

a = bq + r

where a = dividend

b = divisor

q = quotient

r = remainder

=> a - r = bq

$x^4 + x^3 + x^2 - 2x - 6 = [x^2 - 2]q\\q = \frac{x^4 + x^3 + x^2 - 2x - 6}{x^2 - 2}$

Doing long division,

q = $x^2 + x + 3$