Question

When the given roots are α,β,ω,∅,Δ form the polynomial equation.
(Please expand and write the polynomial)

Answer 1

Let α = a,
β = b
ω = c
∅ = d
Δ = e ( for convenience)

We know that,

Polynomial with the above roots is ( x - a) ( x - b) ( x - c) ( x - d) ( x - e)

Let's simplify to get the polynomial.

( x - a) ( x - b) ( x - c) ( x - d) ( x - e)

= (x² -ax-bx+ab)( x - c) ( x - d) ( x - e)

= (x³ - ax² - bx² + abx - cx² + acx + bcx - abc ) ( x - d ) ( x - e )

=[ x³ - (a + b+ c)x² + ( ab+ bc+ca)x - abc ] ( x - d ) ( x - e )

= [x^4 - (a + b+ c)x³ + ( ab+ bc+ca)x²- abcx -dx³ + (a + b + c )dx² - ( ab + bc + ca )dx + abcd ] ( x - e )

= [ x^4 - ( a + b + c + d )x³+ ( ab + bc + ca + ad + bd + cd )x² - ( abd + bcd + cda + abc )x + abcd ] ( x - e )

= [ x^5 - ( a + b + c + d )x^4+ ( ab + bc + ca + ad + bd + cd )x³ - ( abd + bcd + cda + abc )x²+ abcdx - ex^4 + ( a + b + c + d )ex³ - ( ab + bc + ca + ad + bd+ cd)ex² + ( abd + bcd + cda + abc )ex - abcde

= x^5 - ( a + b + c + d + e )x^4 + ( ab + bc + ca + ad + bd + cd + ae + be + ce + de )x³ - ( abd + bcd + cda + abc + abe + bce + cae + ade + bde + cde )x² + ( abcd + abde + bcde + cdae + abce )x - abcde

Or ,If you want in native variables ,


( x - α) ( x - β) ( x - ω) ( x - ∅) ( x - Δ)

=(x² -αx-βx+αβ)( x - ω) ( x - ∅) ( x - Δ)

=(x³ - αx² - βx² + αβx - ωx² + αωx + βωx - αβω ) ( x - ∅ ) ( x - Δ )

=[ x³ - (α + β+ ω)x² + ( αβ+ βω+ωα)x - αβω ] ( x - ∅ ) ( x - Δ )

=[x^4 - (α + β+ ω)x³ + ( αβ+ βω+ωα)x²- αβωx -∅x³ + (α + β + ω )∅x² - ( αβ + βω + ωα )∅x + αβω∅ ] ( x - Δ )

= [ x^4 - ( α + β + ω + ∅ )x³+ ( αβ + βω + ωα + α∅ + β∅ + ω∅ )x² - ( αβ∅ + βω∅ + ω∅α + αβω )x + αβω∅ ] ( x - Δ )

= [ x^5 - ( α + β + ω + ∅ )x^4+ ( αβ + βω + ωα + α∅ + β∅ + ω∅ )x³ - ( αβ∅ + βω∅ + ω∅α + αβω )x²+ αβω∅x - Δx^4 + ( α + β + ω + ∅ )Δx³ - ( αβ + βω + ωα + α∅ + β∅+ ω∅)Δx² + ( αβ∅ + βω∅ + ω∅α + αβω )Δx - αβω∅Δ

= x^5 - ( α + β + ω + ∅ + Δ )x^4 + ( αβ + βω + ωα + α∅ + β∅ + ω∅ + αΔ + βΔ + ωΔ + ∅Δ )x³ - ( αβ∅ + βω∅ + ω∅α + αβω + αβΔ + βωΔ + ωαΔ + α∅Δ + β∅Δ + ω∅Δ )x² + ( αβω∅ + αβ∅Δ + βω∅Δ + ω∅αΔ + αβωΔ )x - αβω∅Δ .


Hope helped !

Answer 2

$\bold{Answer :}$

{ Here is a short method to help you. }

Since, α, β, ω, ∅ and Δ are the zeroes of the required polynomial, by the relation between zeroes and coefficients, we can write the polynomial as

x⁵ - (Σ α) x⁴ + (Σ αβ) x³ - (Σ αβω) x² + (Σ αβω∅) x - αβω∅Δ

= x⁵ - (α + β + ω + ∅ + Δ) x⁴ + (αβ + βω + ωα + α∅ + β∅ + ω∅ + αΔ + βΔ + ωΔ + ∅Δ) x³ - (αβ∅ + βω∅ + ω∅α + αβω + αβΔ + βωΔ + ωαΔ + α∅Δ + β∅Δ + ω∅Δ) x² + (αβω∅ + αβ∅Δ + βω∅Δ + ω∅αΔ + αβωΔ) x - αβω∅Δ

#$\bold{MarkAsBrainliest}$