Algebra 3/4 delves into advanced algebraic concepts, including polynomial functions, rational expressions, and complex numbers.
Just like regular numbers, polynomials can be divided. The two main methods are long division and synthetic division.
Set up the division just as you would with numbers. Divide the leading terms, multiply, subtract, and bring down the next term. Repeat until you can't go further.
A shortcut for dividing by linear factors like \( x - c \). It involves writing coefficients and performing simple operations to find the quotient and remainder.
The theorem states: If you divide a polynomial \( f(x) \) by \( x - c \), the remainder is \( f(c) \).
Polynomial division helps in simplifying expressions and finding factors, while the Remainder Theorem makes evaluating polynomials at specific values super quick.
\[f(x) = (x - c)q(x) + r\]
Dividing \( x^3 - 6x^2 + 11x - 6 \) by \( x - 1 \) using synthetic division gives a remainder of 0.
For \( f(x) = 2x^2 + 3x + 5 \), the remainder when divided by \( x - 2 \) is \( f(2) = 2(4) + 3(2) + 5 = 8 + 6 + 5 = 19 \).
Polynomial division and the Remainder Theorem help break down polynomials and quickly find remainders.