Dividing polynomials is a key skill in algebra. It's like regular division, but with variables and exponents. You'll learn to split up big polynomials into smaller parts, which helps solve complex math problems.

Mastering polynomial division opens doors to understanding advanced math concepts. It's used in calculus, engineering, and computer science. The techniques you'll learn here will be crucial for tackling more complex equations down the road.

Division of Polynomials

Division of monomials

• Divide coefficients and subtract exponents of like bases to divide monomials
  • $\frac{12x^5}{3x^2} = 4x^3$ divides coefficients (12 ÷ 3 = 4) and subtracts exponents ($x^5 ÷ x^2 = x^{5-2} = x^3$)
• Rewrite answer as reciprocal with positive exponent if dividing monomials results in negative exponent
  • $\frac{3x^2}{6x^5} = \frac{1}{2x^3}$ rewrites $\frac{3x^2}{6x^5} = \frac{1}{2x^{5-2}} = \frac{1}{2x^3}$ to eliminate negative exponent

Polynomials divided by monomials

• Divide each term of polynomial by monomial to divide polynomials by monomials
  • $\frac{6x^3 + 9x^2 - 12x}{3x} = 2x^2 + 3x - 4$ divides each term of $6x^3 + 9x^2 - 12x$ by $3x$
• Term remains unchanged in quotient if not divisible by monomial when dividing polynomials by monomials
  • $\frac{4x^3 + 6x^2 + 5}{2x} = 2x^2 + 3x + \frac{5}{2x}$ leaves 5 unchanged as it's not divisible by $2x$
• This process creates a rational expression, which is a fraction of polynomials

Long division for polynomials

1. Arrange polynomials in descending order of degree
2. Divide leading term of dividend by leading term of divisor
3. Multiply result by divisor and subtract from dividend
4. Repeat process until degree of remainder is less than degree of divisor
5. Quotient is result, final difference is remainder

Synthetic division of polynomials

• Shortcut method for dividing polynomials by linear factors

1. Write coefficients of dividend in descending order, followed by vertical line and opposite of constant term in linear factor
2. Bring down leading coefficient, multiply by divisor term, add result to next coefficient
3. Repeat process until all coefficients used
4. Last number is remainder, numbers above line form coefficients of quotient

Interpretation of polynomial division

• Quotient represents result of division, remainder represents value of original function when divisor equals zero
• Quotient determines behavior of original polynomial function for large values of variable
• Remainder determines y-intercept of original function

Remainder and factor theorems

• Remainder Theorem: remainder when polynomial $P(x)$ divided by $x - a$ equals $P(a)$
  • Evaluates polynomial function for given value without calculating entire function
• Factor Theorem: $x - a$ is factor of polynomial $P(x)$ if and only if $P(a) = 0$
  • Determines factors of polynomial function and finds zeros or roots
• These theorems are useful in algebraic division and factorization of polynomials