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[parent] proof of factor theorem due to Fermat (Proof)

Lemma (cf. factor theorem). If the polynomial

$\displaystyle f(x) := a_0x^n\!+\!a_1x^{n-1}\!+\cdots+\!a_{n-1}x\!+\!a_n$
vanishes at $ x = c$, then it is divisible by the difference $ x\!-\!c$, i.e. there is valid the identic equation
$\displaystyle f(x) \equiv (x\!-\!c)q(x)$ (1)

where $ q(x)$ is a polynomial of degree $ n\!-\!1$, beginning with the term $ a_0x^{n-1}$.

The lemma is here proved by using only the properties of the multiplication and addition, not the division.

Proof. If we denote $ x\!-\!c = y$, we may write the given polynomial in the form

$\displaystyle f(x) = a_0(y\!+\!c)^n\!+\!a_1(y\!+\!c)^{n-1}\!+\cdots+\!a_{n-1}(y\!+\!c)\!+\!a_n.$
It's clear that every $ (y\!+\!c)^k$ is a polynomial of degree $ k$ with respect to $ y$, where $ y^k$ has the coefficient 1 and the constant term is $ c^k$. This implies that $ f(x)$ may be written as a polynomial of degree $ n$ with respect to $ y$, where $ y^n$ has the coefficient $ a_0$ and the term independent on $ y$ is equal to $ a_0c^n\!+\!a_1c^{n-1}\!+\cdots+\!a_{n-1}c\!+\!a_n$, i.e. $ f(c)$. So we have
$\displaystyle f(x) = a_0y^n\!+\!b_1y^{n-1}\!+\!b_2y^{n-2}\!+\cdots+\!b_{n-1}y\!+f(c) = f(c)+y\cdot(a_0y^{n-1}\!+\!b_1y^{n-2}\!+\cdots+\!b_{n-1}\!+\!a_n),$
where $ b_1,\,b_2,\,\ldots,\,b_{n-1}$ are certain coefficients. If we return to the indeterminate $ x$ by substituting in the last identic equation $ x\!-\!c$ for $ y$, we get
$\displaystyle f(x) \equiv f(c)+(x\!-\!c)[a_0(x\!-\!c)^{n-1}\!+\!b_1(x\!-\!c)^{n-2}\!+\cdots+\!b_{n-1}].$
When the powers $ (x\!-\!c)^k$ are expanded to polynomials, we see that the expression in the brackets is a polynomial $ q(x)$ of degree $ n\!-\!1$ with respect to $ x$ and with the coefficient $ a_0$ of $ x^{n-1}$. Thus we obtain
$\displaystyle f(x) \equiv f(c)+(x\!-\!c)q(x).$ (2)

This result is true independently on the value of $ c$. If this value is chosen such that $ f(c) = 0$, then (2) reduces to (1), Q. E. D.

Bibliography

1
Ernst Lindelöf: Johdatus korkeampaan analyysiin (`Introduction to Higher Analysis'). Fourth edition. WSOY, Helsinki (1956).



"proof of factor theorem due to Fermat" is owned by pahio.
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Other names:  proof of factor theorem without division

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Cross-references: expression, expanded, powers, indeterminate, implies, coefficient, clear, proof, division, addition, multiplication, properties, degree, identic equation, difference, divisible, vanishes, polynomial, factor theorem

This is version 6 of proof of factor theorem due to Fermat, born on 2006-02-10, modified 2006-09-29.
Object id is 7610, canonical name is ProofOfFactorTheoremDueToFermat.
Accessed 1332 times total.

Classification:
AMS MSC: 12D05 (Field theory and polynomials :: Real and complex fields :: Polynomials: factorization)
 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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