Factor Theorem

Understanding the Factor Theorem

When we divide a polynomial $P(x)$ by $(x - c)$, sometimes the remainder is zero. We know from the Remainder Theorem that if the remainder is zero, then $P(c) = 0$. So, what does it mean if $P(c) = 0$?

The value $c$ that causes $P(c) = 0$ is called a zero or a root of the polynomial $P(x)$. The Factor Theorem explains the close relationship between these zeros and the factors of the polynomial.

Statement of the Factor Theorem

Let $P(x)$ be a polynomial and $c$ be a real number.

$(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$.

This is a two-way statement:

1. If $(x - c)$ is a factor of $P(x)$, then $P(c) = 0$.

   (If a number is perfectly divisible by another number, the remainder must be zero).

2. If $P(c) = 0$, then $(x - c)$ is a factor of $P(x)$.

   (If the value of the polynomial at $x=c$ is zero, it means $(x-c)$ divides the polynomial exactly).

Connection to the Remainder Theorem

The Factor Theorem is actually a special case of the Remainder Theorem. Recall the division algorithm:

And from the Remainder Theorem, we know $S = P(c)$.

• If $(x - c)$ is a factor, it means $P(x)$ is divisible by $(x - c)$. This only happens if the remainder is zero. Thus, $S = P(c) = 0$.
• If $P(c) = 0$, then $S = 0$. The equation becomes $P(x) = (x - c) \cdot H(x) + 0$, or $P(x) = (x - c) \cdot H(x)$. This shows that $(x - c)$ is a factor of $P(x)$.

Using the Factor Theorem to Factor Polynomials

The Factor Theorem is very useful for finding linear factors of a polynomial and then factoring it completely.

General Steps:

1. Find a Zero: Try guessing or using clues (like the sum of coefficients) to find a value $c$ such that $P(c) = 0$.

2. Confirm Factor: If $P(c) = 0$, then according to the Factor Theorem, $(x - c)$ is a factor of $P(x)$.

3. Divide the Polynomial: Use Horner's method or long division to divide $P(x)$ by the factor $(x - c)$ found. The quotient is $H(x)$.

   $$P(x) = (x - c) \cdot H(x)$$

4. Factor the Quotient: If $H(x)$ can still be factored (e.g., if $H(x)$ is a quadratic or cubic polynomial whose roots can be found), repeat the process starting from step 1 on $H(x)$.

5. Complete Factorization: Write $P(x)$ as the product of all the linear factors found.

Factoring a Polynomial

Let $P(x) = x^3 + 2x^2 - 13x + 10$. We notice that the sum of all coefficients and the constant ($1 + 2 - 13 + 10$) is 0. This indicates that $P(1) = 0$.

1. Confirm Zero:

   Calculate $P(1)$.

   $$P(1) = (1)^3 + 2(1)^2 - 13(1) + 10$$

   $$P(1) = 1 + 2 - 13 + 10$$

   $$P(1) = 0$$

2. Confirm Factor:

   Since $P(1) = 0$, $(x - 1)$ is a factor of $P(x)$.

3. Divide Polynomial:

   We use Horner's method to divide $P(x)$ by $(x - 1)$ ($c = 1$).

   $$\begin{array}{c|cccc} 1 & 1 & 2 & -13 & 10 \\ & & 1 & 3 & -10 \\ \hline & 1 & 3 & -10 & \boxed{0} \\ \end{array}$$

   The quotient is $H(x) = 1x^2 + 3x - 10 = x^2 + 3x - 10$. The remainder is 0, as expected.

   So, $P(x) = (x - 1)(x^2 + 3x - 10)$.

4. Factor the Quotient:

   Factor the quadratic polynomial $H(x) = x^2 + 3x - 10$.

   $$x^2 + 3x - 10 = (x - 2)(x + 5)$$

5. Complete Factorization:

   Combine all factors.

   $$P(x) = (x - 1)(x - 2)(x + 5)$$

Exercise

Let $P(x) = x^3 - 2x^2 - 21x - 18$. Show that $P(-1) = 0$, and use this to factor $P(x)$ completely.

Answer Key

1. Show $P(-1) = 0$:

   $$P(-1) = (-1)^3 - 2(-1)^2 - 21(-1) - 18$$

   $$P(-1) = -1 - 2(1) + 21 - 18$$

   $$P(-1) = -1 - 2 + 21 - 18$$

   $$P(-1) = -3 + 3 = 0$$

   Proven $P(-1) = 0$.

2. Confirm Factor:

   Since $P(-1) = 0$, $(x - (-1)) = (x + 1)$ is a factor of $P(x)$.

3. Divide Polynomial (Horner's Method with $c = -1$):

   $$\begin{array}{c|cccc} -1 & 1 & -2 & -21 & -18 \\ & & -1 & 3 & 18 \\ \hline & 1 & -3 & -18 & \boxed{0} \\ \end{array}$$

   The quotient is $H(x) = x^2 - 3x - 18$.

   So, $P(x) = (x + 1)(x^2 - 3x - 18)$.

4. Factor the Quotient:

   Factor $H(x) = x^2 - 3x - 18$.

   $$x^2 - 3x - 18 = (x - 6)(x + 3)$$

5. Complete Factorization:

   $$P(x) = (x + 1)(x - 6)(x + 3)$$