Factor Theorem. Remainder Theorem When a function f(x), is divided by x – k the remainder is f(k) Example 1.

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Presentation transcript:

Factor Theorem

Remainder Theorem When a function f(x), is divided by x – k the remainder is f(k) Example 1

Example 2

Factor Theorem What happens when a remainder is equal to zero? Let’s revisit the division statement from the division we did earlier. If the remainder is zero, then the expression would be factored.

Factor Theorem If we can find a value ‘k’, that makes the expression f(x) equal to zero then, (x - k) will be a factor of f(x). Then we can use division to find the other factors. Now the trick is to find the value of ‘k’. To see how to do this we consider a function that is in factored form already. Notice that the constant term of each factor multiply together to give the constant term.

Factor Property To find the value of ‘k’ that will make f(k)=0, try factors of the constant term of f(x). Example 2: Factor the following polynomial. Try factors of 24 to find ‘k’. Therefore (x +2) is a factor of f(x).

Divide to find the other factors. Divide f(x) by (x + 2) to find the other factors. How many times does x go into x 3 ? Multiply x+2 by x 2. Subtract down. Note 4x 2 – (-3x 2 )=4x 2 + (+3x 2 )=7x 2 Bring down +2x. How many times does x go into 7x 2 ? Multiply x-3 by 7x. Subtract down. Bring down –5. How many times does x go into 23x ? Multiply x-3 by 23. Subtract down.

Continue to factor So we can write

Example 3 Solution:

Example 3 Continued