Remainder and factor theorem

REMAINDER AND FACTOR THEOREM     

   PRACTICE SUMS 👈

1. Remainder Theorem:

Statement:

If a polynomial $f(x)$ is divided by $(x - a)$, then the remainder is equal to $f(a)$

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🔹 How to Use:

• Take the polynomial $f(x)$
  

• Substitute $x = a$
  

• The result is the remainder

Example:
If $f(x) = x^3 - 4x + 2$ find the remainder when divided by $x - 2$:
👉 $f(2) = (2)^3 - 4(2) + 2 = 8 - 8 + 2 = 2$
✅ Remainder = 2

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✅ 2. Factor Theorem:

Statement:

If $f(a) = 0$ then $(x - a)$ is a factor of the polynomial $f(x)$

 

Conversely, if $(x - a)$ is a factor of $f(x)$ then $f(a) = 0$

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🔹 How to Use:

• Substitute $x = a$ into $f(x)$
  

• If $f(a) = 0$ then $(x - a)$
  

Example:
Let $f(x) = x^2 - 5x + 6$      Is $(x - 2)$ a factor?

👉 Check $f(2)=2^2-5(2)+6=4-10+6=0$

✅ So, (x - 2) is a factor