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PROOF 102

Prove.

The relation between roots and coefficients

On the real numbers ℝ.

What you may use

A1f(α)=0 ⟹ f=(x−α)(px+d)
factor theorem and degree (Proof 24)
A2equal at every x ⟹ equal coefficients
equality of polynomials
A3f(x)=g(x) ⟹ f(t)=g(t)
substitution
A4(fg)(t) = f(t)g(t)
value of a product
A5(f±g)(t) = f(t)±g(t)
value of a sum and difference
A6x(t) = t, c(t) = c
values of x and constants
A7a≠0, b≠0 ⟹ ab ≠ 0
no zero divisors
A8a(b + c) = ab + ac
distributive law
A9(ab)c = a(bc)
associativity of multiplication
A10ab = ba
commutativity of multiplication
A11a × 1 = a
multiplicative identity
A12a ≠ 0 ⟹ a·a⁻¹ = 1
multiplicative inverse
A13(a+b)+c = a+(b+c)
associativity of addition
A14a + b = b + a
commutativity of addition
A15a + 0 = a
additive identity
A16a + (−a) = 0
additive inverse
A17t × 0 = 0
lemma of Proof 01
A18(−a)b = −(ab)
product with a negative (Proof 01)
A19(−a) + (−b) = −(a+b)
sum of inverses (Proof 80)
A20x+y=0, x+z=0 ⟹ y=z
uniqueness of additive inverses (Proof 80)
A21(−a)(−b) = ab
product of negatives (Proof 01)
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

A root of ax² + bx + c = 0 (a ≠ 0) is a number x for which the equation holds upon substitution.

Proof.

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