Remainder Theorem
Apply the Remainder Theorem to evaluate polynomials quickly: dividing p(x) by (x-a) gives remainder p(a), turning synthetic division into a fast function evaluation tool.
Key Concepts
If a polynomial $f(x)$ is divided by $x k$, the remainder is $r = f(k)$.
Example 1: To find the remainder of $f(x) = x^3 2x^2 5x + 6$ when divided by $x 3$, we know the remainder is $f(3)$. \n $f(3) = (3)^3 2(3)^2 5(3) + 6 = 27 18 15 + 6 = 0$. The remainder is $0$. \n Example 2: The remainder when $P(x) = 2x^3 7x + 1$ is divided by $x+2$ is $P( 2)$. \n $P( 2) = 2( 2)^3 7( 2) + 1 = 16 + 14 + 1 = 1$. The remainder is $ 1$.
The Remainder Theorem is a fantastic mathematical shortcut that connects division to evaluating functions. It tells us that the leftover 'remainder' from a synthetic division problem is the exact same number you'd get if you plugged the divisor's value into the polynomial. So, dividing $f(x)$ by $x 3$ gives a remainder that equals $f(3)$. Cool, right?
Common Questions
What does the Remainder Theorem state?
The Remainder Theorem states that when a polynomial p(x) is divided by (x-a), the remainder equals p(a). You can evaluate a polynomial at any value by performing synthetic division and reading the last number in the bottom row.
How is the Remainder Theorem different from the Factor Theorem?
The Remainder Theorem gives the remainder when dividing by (x-a). The Factor Theorem is a special case: if that remainder is zero, meaning p(a)=0, then (x-a) is a factor of the polynomial.
How do you use synthetic division with the Remainder Theorem?
Write the coefficients of p(x), use the value a as the divisor, and carry out synthetic division. The final value in the bottom row is the remainder, which equals p(a) without substituting into the full polynomial.