Key Concepts

• Given two polynomials, where the divisor is in the form $x-k$, how to use synthetic division to divide

  1. Write k for the divisor.
  2. Write the coefficients of the dividend.
  3. Bring the lead coefficient down.
  4. Multiply the lead coefficient by k. Write the product in the next column.
  5. Add the terms of the second column.
  6. Multiply the result by k. Write the product in the next column.
  7. Repeat steps $5$ and $6$ for the remaining columns.
  8. Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree $0$. The next number from the right has degree $1$, and the next number from the right has degree $2$, and so on.

• Equivalent Fractions Property
  • If $a,b,c$ are whole numbers where $b\ne 0,c\ne 0$, then
    ${\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}$ and ${\Large\frac{a\cdot c}{b\cdot c}}={\Large\frac{a}{b}}$
• Zero Exponent
  • If $a$ is a non-zero number, then ${a}^{0}=1$.
  • Any nonzero number raised to the zero power is $1$.
• Quotient Property for Exponents
  • If $a$ is a real number, $a\ne 0$, and $m,n$ are whole numbers, then
    ${\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n$ and ${\Large\frac{{a}^{m}}{{a}^{n}}}={\Large\frac{1}{{a}^{n-m}}},n>m$
• Quotient to a Power Property for Exponents
  • If $a$ and $b$ are real numbers, $b\ne 0$, and $m$ is a counting number, then
    ${\Large{\left(\frac{a}{b}\right)}}^{m}={\Large\frac{{a}^{m}}{{b}^{m}}}$
  • To raise a fraction to a power, raise the numerator and denominator to that power.

Glossary

Synthetic Division   Synthetic division is a shortcut that can be used when the divisor is a binomial in the form  $x–k$, for a real number $k$. In synthetic division, only the coefficients are used in the division process.