can anyone tell me Factor Theorem plzz clearly it is too hard
Answers
yes , i can tell u it but I cant clear your concept u have to do it by urs ok
Step-by-step explanation:
The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm tells us
\displaystyle f\left(x\right)=\left(x-k\right)q\left(x\right)+rf(x)=(x−k)q(x)+r.
If k is a zero, then the remainder r is \displaystyle f\left(k\right)=0f(k)=0 and \displaystyle f\left(x\right)=\left(x-k\right)q\left(x\right)+0f(x)=(x−k)q(x)+0 or \displaystyle f\left(x\right)=\left(x-k\right)q\left(x\right)f(x)=(x−k)q(x).
Notice, written in this form, x – k is a factor of \displaystyle f\left(x\right)f(x). We can conclude if k is a zero of \displaystyle f\left(x\right)f(x), then \displaystyle x-kx−k is a factor of \displaystyle f\left(x\right)f(x).
Similarly, if \displaystyle x-kx−k is a factor of \displaystyle f\left(x\right)f(x), then the remainder of the Division Algorithm \displaystyle f\left(x\right)=\left(x-k\right)q\left(x\right)+rf(x)=(x−k)q(x)+r is 0. This tells us that k is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree n in the complex number system will have n zeros. We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.
A GENERAL NOTE: THE FACTOR THEOREM
According to the Factor Theorem, k is a zero of \displaystyle f\left(x\right)f(x) if and only if \displaystyle \left(x-k\right)(x−k) is a factor of \displaystyle f\left(x\right)f(x).
HOW TO: GIVEN A FACTOR AND A THIRD-DEGREE POLYNOMIAL, USE THE FACTOR THEOREM TO FACTOR THE POLYNOMIAL.
Use synthetic division to divide the polynomial by \displaystyle \left(x-k\right)(x−k).
Confirm that the remainder is 0.
Write the polynomial as the product of \displaystyle \left(x-k\right)(x−k) and the quadratic quotient.
If possible, factor the quadratic.
Write the polynomial as the product of factors.
hope it will help u ☺️
Answer:
The remainder theorem tells us that for any polynomial f(x) , if you divide it by the binomial x−a , the remainder is equal to the value of f(a) . The factor theorem tells us that if a is a zero of a polynomial f(x) , then (x−a) is a factor of f(x) , and vice-versa.
Step-by-step explanation:
Example:
Examine whether x + 2 is a factor of x3 + 3x2 + 5x + 6.
Solution: To begin with, we know that the zero of the polynomial (x + 2) is –2. Let p(x) = x3 + 3x2 + 5x + 6
Then, p(–2) = (–2)3 + 3(–2)2 + 5(–2) + 6 = –8 + 12 – 10 + 6 = 0
According to the factor theorem, if p(a) = 0, then (x – a) is a factor of p(x). In this example, p(a) = p(- 2) = 0
Therefore, (x – a) = {x – (-2)} = (x + 2) is a factor of ‘x3 + 3x2 + 5x + 6’ or p(x).