6.10: Synthetic Division of Polynomials
The volume of a rectangular prism is
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James Sousa: Polynomial Division: Synthetic Division
Guidance
Synthetic division is an alternative to long division from the previous concept. It can also be used to divide a polynomial by a possible factor,
Example A
Divide
Solution: Using synthetic division, the setup is as follows:
To “read” the answer, use the numbers as follows:
Therefore, 2 is a solution, because the remainder is zero. The factored polynomial is
Example B
Determine if 4 is a solution to
Using synthetic division, we have:
The remainder is 304, so 4 is not a solution. Notice if we substitute in
Remainder Theorem: If
This means that if you substitute in
Example C
Determine if
Solution: If you use synthetic division, the factor is not in the form
This means that
Intro Problem Revisit
If
If we use synthetic division, the factor is not in the form
When we perform the synthetic division, we get a remainder of 0. This means that
Guided Practice
1. Divide
2. Divide
3. Is 6 a solution for
Answers
1. Using synthetic division, divide by 3.
The answer is
2. Using synthetic division, divide by
The answer is
3. Put a zero placeholder for the
The resulting polynomial is
The solutions to this polynomial are 6,
Vocabulary
 Synthetic Division

An alternative to long division for dividing
f(x) byk where only the coefficients of \begin{align*}f(x)\end{align*}f(x) are used.
 Remainder Theorem
 If \begin{align*}f(k) = r\end{align*}, then \begin{align*}r\end{align*} is also the remainder when dividing by \begin{align*}(x  k)\end{align*}.
Practice
Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
 \begin{align*}(x^3+6x^2+7x+10) \div (x+2)\end{align*}
 \begin{align*}(4x^315x^2120x128) \div (x8)\end{align*}
 \begin{align*}(4x^25) \div (2x+1)\end{align*}
 \begin{align*}(2x^415x^330x^220x+42) \div (x+9)\end{align*}
 \begin{align*}(x^33x^211x+5) \div (x5)\end{align*}
 \begin{align*}(3x^5+4x^3x2) \div (x1)\end{align*}
 Which of the division problems above generate no remainder? What does that mean?
 What is the difference between a zero and a factor?
 Find \begin{align*}f(2)\end{align*} if \begin{align*}f(x)=2x^45x^310x^2+21x4\end{align*}.
 Now, divide \begin{align*}2x^45x^310x^2+21x4\end{align*} by \begin{align*}(x + 2)\end{align*} synthetically. What do you notice?
Find all real zeros of the following polynomials, given one zero.
 \begin{align*}12x^3+76x^2+107x20; 4\end{align*}
 \begin{align*}x^35x^22x+10; 2\end{align*}
 \begin{align*}6x^317x^2+11x2; 2\end{align*}
Find all real zeros of the following polynomials, given two zeros.
 \begin{align*}x^4+7x^3+6x^232x32; 4, 1\end{align*}
 \begin{align*}6x^4+19x^3+11x6; 0, 2\end{align*}
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Term  Definition 

Oblique Asymptote  An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division. 
Oblique Asymptotes  An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division. 
Remainder Theorem  The remainder theorem states that if , then is the remainder when dividing by . 
Synthetic Division  Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used. 
Image Attributions
Here you'll learn how to use synthetic division as a shortcut and alternative to long division (in certain cases) and to find zeros.