Question

State factor theorem.
​

Answer 1

Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.

Use synthetic division to divide the polynomial by (x−k).

Confirm that the remainder is 0.

Write the polynomial as the product of (x−k) and the quadratic quotient.

If possible, factor the quadratic.

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).

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