Binomial Theorem

• The process of writing a polynomial as a product is called factoring

• If a polynomial does not factor using integer coefficients, then it is prime or irreducible over the integers

• A polynomial is completely factored when each of its factors is prime

• Binomial Formula $(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^k{b}^{n-k}$

  • $(1+x{)}^n={\sum }_{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k$
  • Perfect Square Trinomial: square of sum/difference
    • $(a+b{)}^2=a^2+2ab+b^2=(-a-b{)}^2$
      • $(Ax+B{)}^2=(Ax{)}^2+2ABx+B^2$
    • $(a-b{)}^2=a^2-2ab+b^2$
  • cube of sum/difference
    • $(a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3$
    • $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$
  • Difference of two squares $a^2-b^2=(a-b)(a+b)$
  • Sum of squares $a^2+b^2=(a+b{)}^2-2ab=(a-b{)}^2+2ab$
  • Sum/difference of cubes (SOAP)
    • $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$