Question

when a polynomial f (x) is divided by xsquare -5 the quotient is xsquare -2x-3 and remainder is zero find the polynomial and all its zeroes

Answer 1

we know that P(x) = q(x)×g(x)+r(x)
p(x) = f(x) = ?
q(x) = x²-2x-3
g(x) = x²-5
r(x) = 0

1 ]
f(x) = q(x)×g(x)+r(x)

f(x) = (x²-2x-3)×(x²-5)+0

f(x) = x²(x²-2x-3)-5(x²-2x-3)

f(x) = x⁴-2x³-3x²-5x²+10x+15

f(x) = x⁴-2x³-8x²+10x+15


2 ]
x⁴-2x³-8x²+10x+15 = 0

(x+√5) • (x- √5) • (x + 1) • (x - 3) = 0

so zeroes are

x = -√5

x = +√5

x = -1

x = 3

Answer 2

hello....

f(x) = ?

g(x) = x2 - 5

q(x) = x2 - 2x - 3

r(x) = 0

By division algorithm for polynomials, we have

f(x) = q(x) . g(x) + r(x)

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

f(x) = x4 - 2x3 - 3x2 - 5x2 + 10x + 15

f(x) = x­4 - 2x3 - 8x2 + 10x + 15

So, the required polynomial is f(x) = x­4 - 2x3 - 8x2 + 10x + 15

Now,

q(x) and g(x) will be factors of f(x)

x2 - 5 = 0 and x2 - 2x - 3 = 0

x2 - (√5)2 = 0 and x2 + x - 3x - 3 = 0

(x - √5)(x + √5) = 0 and (x + 1)(x - 3) = 0

x = √5, x = -√5, x = -1 and x = 3

So, the zeroes are α = √5, β = -√5, γ = -1 and δ = 3

thank you...

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