Question

polynomial p(x) in p(2)=0 then p(x) factor is​

Answer 1

Step-by-step explanation:

0 is the factor.............

Answer 2

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Factor theorem is a special case for the remainder theorem. It states that when the polynomial p(x) is equally divided by another polynomial g(x) with the divisor x-a and if the remainder is zero i.e. R(x) = 0. So x-a is said to be the factor of that polynomial p(x). We can write it in the form: P(x) = (x-a).q(x) where the remainder is said to be zero.

This theorem also shows us the relationship between factors of the polynomials as well as zero of polynomial and helps to find out the zero of a polynomial having two or higher degrees.

Definition

If p(x) is a polynomial of degree n which is greater than or equal to one and a is any real number which will be the divisor, then there will be two conditions fulfilled:

If p (a) =0, then x-a is a factor of that polynomial p(x).

x-a would be the factor of the polynomial if the r(x) i.e. remainder is 0.

Proof

By the Remainder Theorem, Dividend = Divisor * Quotient + Remainder where P(x) = (x – a). Q(x) + p (a)

Consider a polynomial p(x) which is divided by (x-a), then p (a) =0. Therefore, p(x) = (x-a).q(x) +p (a) = (x-a).q(x) +0 = (x-a).q(x). Thus, x-a is a factor of p(x) when the remainder is zero.

If the (x-a) is a factor of polynomial p(x), then the remainder must be zero. So, we can say that x-a exactly divides p(x). Thus p(x) =0. Hence, the theorem is being proved.

Example 1: Examine whether x – 2 is a factor of

x^{2}-7x+10

.

By putting the value of P(2) in the equation

we get P(2) =

= 4-14+10 = 0

Since P(2) = 0. Therefore, x – 2 is a factor of P(x).