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Remainder Theorem:

If is a polynomial in , and is divided by ; then the remainder is .

So for example, If is divided by ; then the remainder is . All you have to do is to substitute and calculate the value of the function.

Factor Theorem:

When a polynomial is divided by ; then the remainder is . And if the remainder , then is a factor of the polynomial

Let’s do a couple of examples to make it more clear.

Example 1:

Find the value of if the division of by leaves a remainder of .

Answer:

It is given that the remainder is .

Therefore, if you substitute , then the remainder should be .

Hence

Example 2:

If is a factor of , then find the value of .

Answer:

Since is a factor, therefore when we substitute into the polynomial, the remainder would be .

Therefore

Example 3:

Find such that the polynomial has as its factors. Once you find the value of the , factorize the polynomial.

Answer:

is a factor of the given polynomial,

Similarly,

is a factor of the given polynomial,

On solving (i) and (ii) simultaneously, we get .

Then the polynomial becomes the following:

To factorize:

Step 1: Divide the polynomial by as shown

Therefore

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