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Discover how polynomial roots connect to factors. Use the Factor Theorem to find zeros, identify linear factors, and factor polynomials.
---
## 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 $$P(c) = 0$$?
Visible text: When we divide a polynomial by , sometimes the remainder is zero. We know from the Remainder Theorem that if the remainder is zero, then . So, what does it mean if ?
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.
Visible text: The value that causes is called a **zero** or a **root** of the polynomial . 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.
Visible text: Let be a polynomial and be a real number.
$$(x - c)$$ is a **factor** of $$P(x)$$ **if and only if** $$P(c) = 0$$.
Visible text: is a **factor** of **if and only if** .
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).
Visible text: 1. **If is a factor of , then .**
(If a number is perfectly divisible by another number, the remainder must be zero).
2. **If , then is a factor of .**
(If the value of the polynomial at is zero, it means 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:
```math
P(x) = (x - c) \cdot H(x) + S
```
And from the Remainder Theorem, we know $$S = P(c)$$.
Visible text: And from the Remainder Theorem, we know .
```math
P(x) = (x - c) \cdot H(x) + 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 $$P(x)$$.
Visible text: - If is a factor, it means is divisible by . This only happens if the remainder is zero. Thus, .
- If , then . The equation becomes , or . This shows that is a factor of .
## 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 $$H(x)$$.
```math
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.
Visible text: 1. **Find a Zero:** Try guessing or using clues (like the sum of coefficients) to find a value such that .
2. **Confirm Factor:** If , then according to the Factor Theorem, is a factor of .
3. **Divide the Polynomial:** Use Horner's method or long division to divide by the factor found. The quotient is .
4. **Factor the Quotient:** If can still be factored (e.g., if is a quadratic or cubic polynomial whose roots can be found), repeat the process starting from step on .
5. **Complete Factorization:** Write 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 $$P(1) = 0$$.
Visible text: Let . We notice that the sum of all coefficients and the constant () is . This indicates that .
1. **Confirm Zero:**
Calculate $$P(1)$$.
```math
P(1) = (1)^3 + 2(1)^2 - 13(1) + 10
```
```math
P(1) = 1 + 2 - 13 + 10
```
```math
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 $$(x - 1)$$ ($$c = 1$$).
```math
\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 $$H(x) = x^2 + 3x - 10$$.
```math
x^2 + 3x - 10 = (x - 2)(x + 5)
```
5. **Complete Factorization:**
Combine all factors.
```math
P(x) = (x - 1)(x - 2)(x + 5)
```
Visible text: 1. **Confirm Zero:**
Calculate .
2. **Confirm Factor:**
Since , is a factor of .
3. **Divide Polynomial:**
We use Horner's method to divide by ().
The quotient is . The remainder is , as expected.
So, .
4. **Factor the Quotient:**
Factor the quadratic polynomial .
5. **Complete Factorization:**
Combine all factors.
## Exercise
Let $$P(x) = x^3 - 2x^2 - 21x - 18$$. Show that $$P(-1) = 0$$, and use this to factor $$P(x)$$ completely.
Visible text: Let . Show that , and use this to factor completely.
### Answer Key
1. **Show $$P(-1) = 0$$:**
```math
P(-1) = (-1)^3 - 2(-1)^2 - 21(-1) - 18
```
```math
P(-1) = -1 - 2(1) + 21 - 18
```
```math
P(-1) = -1 - 2 + 21 - 18
```
```math
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 $$c = -1$$):**
```math
\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 $$H(x) = x^2 - 3x - 18$$.
```math
x^2 - 3x - 18 = (x - 6)(x + 3)
```
5. **Complete Factorization:**
```math
P(x) = (x + 1)(x - 6)(x + 3)
```
Visible text: 1. **Show :**
Proven .
2. **Confirm Factor:**
Since , is a factor of .
3. **Divide Polynomial (Horner's Method with ):**
The quotient is .
So, .
4. **Factor the Quotient:**
Factor .
5. **Complete Factorization:**