Question

what is meant by factor theorem

Answer 1

Let p(x) be a polynomial of a degree greater than or equal to one and a be a real number such that p(a) = 0

Then , we have to show that ( x-a ) is a factor of p(x).

Let q(x) be the quotient when p(x) is divisible by ( x-a)

By remainder theorem

Dividend = Divisor x Quotient + Remainder

p(x) = ( x-a ) x q(x) + p(a) [Remainder theorem]

p(x) = (x-a) x q(x)  [p(a) = 0 ]

(x-a) is a factor of p(x)

conversely ,

Let (x-a) be a factor of p(x) . Then we have to prove that p(a) = 0

Now

(x-a) is a factor of p(x)

p(x) , when divided by (x-a) gives Remainder zero . But by the remainder theorem , p(x) when divided by (x-a) gives the remainder equal to p(a)

.: p(a) = 0

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Answer 2

$\huge\boxed{\fcolorbox{lime}{yellow}{⭐ANSWER⤵࿐}}$

$\sf\large\underline\green{✴Introduction}$

In mathematics, factor theorem is used when factoring the polynomials completely. It is a theorem that links factors and zeros of the polynomial.

According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then, (x-a) is a factor of f(x), if f(a)=0.

Also, we can say, if (x-a) is a factor of polynomial f(x), then f(a) = 0. This proves the converse of the theorem. Let us see the proof of this theorem along with examples.

$\sf\large\underline\red{⭐Factor Theorem :-}$

Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial. It is a special case of a polynomial remainder theorem.

As discussed in the introduction, a polynomial f(x) has a factor (x-a), if and only if, f(a) = 0. It is one of the methods to do the factorisation of a polynomial.

$\sf\large\underline\blue{✴Proof:-}$

Here we will prove the factor theorem, according to which we can factorise the polynomial.

Consider a polynomial f(x) which is divided by (x-c), then f(c)=0.

Using remainder theorem,

➡f(x)= (x-c)q(x)+f(c)

Where f(x) is the target polynomial and q(x) is the quotient polynomial.

➡Since, f(c) = 0, hence,

➡f(x)= (x-c)q(x)+f(c)

➡f(x) = (x-c)q(x)+0

➡f(x) = (x-c)q(x)

Therefore, (x-c) is a factor of the polynomial f(x).

$\sf\large\underline\red{⭐Hope It's Help You !!✔ }$