Advanced algebraic concepts including polynomials, rational expressions, and complex numbers.
Dividing polynomials works a lot like dividing regular numbers, just with variables and exponents. You can use long division or synthetic division (a shortcut for certain cases).
If you divide a polynomial \( f(x) \) by \( x - a \), the remainder is \( f(a) \). This makes evaluating polynomials super quick!
Division helps us rewrite complicated expressions and understand how polynomials behave, especially when finding zeros.
\[f(x) = (x - a)Q(x) + R\]
Dividing \( x^3 + 2x^2 - 5x + 6 \) by \( x - 2 \) gives a remainder of \( f(2) = 8 + 8 - 10 + 6 = 12 \).
Using synthetic division to divide \( x^2 - 4 \) by \( x - 2 \) gives a quotient of \( x + 2 \) and remainder 0.
Polynomial division breaks big expressions into simpler parts, and the Remainder Theorem gives a shortcut for finding remainders.