Dividing Polynomials

Polynomial Long Division

Suppose we would like to divide one polynomial by another. The procedure is similar to long division of numbers and is illustrated in the following example:

Example 1

Divide Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-2x-15} (the dividend or numerator) by $x+3$ (the divisor or denominator)}} Similar to long division of numbers, we set up our problem as follows:

{\begin{array}{rl}\\x+3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x^{2}-2x-15\end{array}}\end{array}}

First we have to answer the question, how many times does Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+3} go into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} ? To find out, divide the leading term of the dividend by leading term of the divisor. So it goes in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} times. We record this above the leading term of the dividend:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rl}&~~\,x\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ \end{array}}

, and we multiply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+3} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and write this below the dividend as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rl}&~~\,x\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ &\!\!\!\!-\underline{(x^2+3x)~~~}\\ \end{array}}

Now we perform the subtraction, bringing down any terms in the dividend that aren't matched in our subtrahend:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rl}&~~\,x\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ &\!\!\!\!-\underline{(x^2+3x)~~~}\\ &\!\!\!\!~~~~~~-5x-15~~~\\ \end{array}}

Now we repeat, treating the bottom line as our new dividend:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rl}&~~\,x-5\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ &\!\!\!\!-\underline{(x^2+3x)~~~}\\ &\!\!\!\!~~~~~~-5x-15~~~\\ &\!\!\!\!~~~-\underline{(-5x-15)~~~}\\ &\!\!\!\!~~~~~~~~~~~~~~~~~~~0~~~\\ \end{array}}

In this case we have no remainder.

Example 2

What about a non-divisible polynomials like this?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3x^2 + 3x - 4) / (x - 4)}

Long division method
1	We first consider the highest-degree terms from both the dividend and divisor, the result is the first term of our quotient.	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3x^2) / (x) = 3x}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{r\|ccc} x - 4 & 3x^2 & 3x & -4 \\ \hline \hline & 3x^2 & 3x & \\ 3x(x - 4) & 3x^2 & -12x & \\ \hline & & 15x & -4 \\ 15(x - 4) & & 15x & -60 \\ \hline & & & 56 \\ \end{array}}
2	Then we multiply this by our divisor.	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3x) \times (x - 4) = 3x^2 - 12x}
3	And subtract the result from our dividend.	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3x^2 + 3x - 4) - (3x^2 - 12x) = 15x - 4}
4	Now once again with the highest-degree terms of the remaining polynomial, and we got the second term of our quotient.	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (15x) / (x) = 15}
5	Multiplying...	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (15) \times (x - 4) = 15x - 60}
6	Subtracting...	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (15x - 4) - (15x - 60) = 56}
7	We are left with a constant term - our remainder:	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcr} Q(x) = 3x + 15 & & R = 56 \end{array}}

So finally:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3x^2 + 3x - 4) = (3x + 15) \times (x - 4) + 56}

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