Factor Theorem

Factor Theorem - How it Works - Video

Division Algorithm for Polynomials

Division Algorithm for Polynomials:

This algorithm works for any polynomial. We can use long division to find the dividend, divisor, quotient, and remainder. Or we can use synthetic division.

If we have the example of 5 divided by 3, 5 is the dividend, 3 is the divisor, 1 is the quotient, and 2 is the remainder. We can write that in two different ways. 5/3 = 1 + 2/3 or 5 = 1 * 3 + 2. Each has their own advantages or disadvantages. 

Remainder and Factor Theorems

Remainder and Factor Theorems:

The remainder theorem is f(x) = (x - c) * q(x) + f(c), where f(c) is NOT 0.

The remainder theorem states if any polynomial is divide by an expression, x - c, then the remainder is f(c).

Remember, dividend = divisor * quotient + remainder => f(x) = p(x) * q(x) + r(x). So when we divide by f(x) by (x - c), we are left with f(c) or r(x).

The factor theorem is f(x) = (x - c) * q(x) + f(c), where f(c) is 0.

The factor theorem states if any polynomial is divided by an expression, (x - c), and if and only if the remainder is f(c) = 0.

Remember, dividend = divisor * quotient + remainder => f(x) = p(x) * q(x) + r(x). So when we divide by f(x) by (x - c), we are left 0 for f(c) or r(x).

Example 1 - Long Division

Example 1 - Long Division:

Here we used long division to find the remainder. In this case, it is 0. Since the remainder is 0, it follows the Factor Theorem and it shows that x - 4 is a factor of f(x) = x3 - 4x2 - 3x +12. This is means we can write the function as f(x) = (x - 4) * (x2 - 3) + 0.

Here are more examples of long division.

Example 1 - Substitution

Example 1 - Substitution:

The quicker way to find if an expression is a factor is to find c and substitute that value into the function.

Our function is f(x) = x3 - 4x2 - 3x +12 and we are dividing by x - 4. To find c, we need to see if it follows the form x - c. In this case it does so c = 4. If the expression were x + 5, then c would be -5 since we can transform x + 5 to x - (-5).

So this is also shows that x - 4 is a factor since the output is 0. 

If the output is not 0, then the expression is not a factor.

Since f(4) = 0, (x - 4) is a factor of our function f(x) = x3 - 4x2 - 3x + 12, because the output is 0.

Let's see what happens if we had (x + 3) as our divisor, our c value would be -3 because (x + 3) ⇔ (x - [-3]) ⇔ (x - c), c = -3. 

Since f(-3) = -42, (x + 3) is not a factor of our function f(x) = x3 - 4x2 - 3x + 12, because the output is not 0.

Live Worksheet

Teacher - Edpuzzle Link