If a polynomial \$f(x)\$ is divided by \$(x - a)\$, the remainder is \$f(a)\$.
The Remainder Theoremprovides a quick way to find the remainder without performing the full division.
Using the Remainder Theorem
- Identify the divisor \$(x - a)\$.
- Substitute \$a\$ into the polynomial \$f(x)\$.
- The result is the remainder.
Find the remainder when \$f(x) = x^4 - 2x^3 + x - 1\$ is divided by \$x + 1\$.
The Factor Theorem
A polynomial \$f(x)\$ has a factor \$(x - a)\$ if and only if \$f(a) = 0\$.
The Factor Theoremis an extension of the Remainder Theorem. It tells us that if the remainder is \$0\$, then the divisor is a factor of the polynomial.
Using the Factor Theorem
- Identify the divisor \$(x - a)\$.
- Substitute \$a\$ into the polynomial \$f(x)\$.
- If \$f(a) = 0\$, then \$(x - a)\$ is a factor.
Check if \$x + 2\$ is a factor of \$f(x) = x^3 + 4x^2 + 4x + 8\$.
Be careful with the sign in the divisor. If the divisor is \$x + a\$, substitute \$-a\$ into the polynomial.
How do the Remainder and Factor Theorems illustrate the power of mathematical shortcuts? What are the limitations of these theorems?
