document.write( "Question 691232: I need help figuring out this problem. What is the polynomial function? 13i and 5-10i \n" ); document.write( "

Algebra.Com's Answer #426589 by fcabanski(1391) $\"\"$ $\"About$

You can put this solution on YOUR website!
The key is recognizing that complex roots are generated in pairs. So the roots are 13i, -13i, 5-10i, 5+10i. In each case x= the root. Subtract the root from both sides to find the factor.

\n" ); document.write( "x=13i so x-13i is the factor. x=-13i so x+13i is the factor. x=5-10i s0 x-5+10i is the factor. x=5+10i so x-5-10i is the factor.

\n" ); document.write( "You can multiply the factors together in any way, but generally it's best to multiply the factors based on the like roots together. (x-13i)(x+13i)(x-5+10i)(x-5-10i). Usually this solution is sufficient. But you can multiply the factors.

\n" ); document.write( "(x-13i)(x+13i) = x^2 +169. The detailed explanation is after the problem solution. Remember i^2 = -1.

\n" ); document.write( "(x-5+10i)(x-5-10i)=x^2 -10x +125

\n" ); document.write( "The two factors remaining are those answers (x^2 +169)(x^2 -10x +125)= x^4 -10x^3 +294x^2 -1690x +21125
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Hope the solution helped. Sometimes you need more than a solution. Contact fcabanski@hotmail.com for online, private tutoring, or personalized problem solving (quick for groups of problems.)\r
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Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:

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\n" ); document.write( " YOUR ANSWER\n" ); document.write( "

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Graphical form: $\"%28x-13i%29%2A%28x%2B13i%29\"$ simplifies to $\"x%5E2-169%2Ai%5E2\"$
Text form: (x-13i)*(x+13i) simplifies to x^2-169*i^2
Cartoon (animation) form: $\"simplify_cartoon%28+%28x-13i%29%2A%28x%2B13i%29+%29\"$
For tutors: simplify_cartoon( (x-13i)*(x+13i) )
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If you have a website, here's a link to this solution.
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DETAILED EXPLANATION
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Look at $\"highlight_red%28+%28x-13%2Ai%29%2A%28x%2B13%2Ai%29+%29\"$ .
Expanded term $\"%28x%2B13%2Ai%29\"$ by using associative property on $\"%28x-13%2Ai%29\"$
It becomes $\"highlight_green%28+%28x%2B13%2Ai%29%2Ax+%29-highlight_green%28+%28x%2B13%2Ai%29%2A13%2Ai+%29\"$ .
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Look at $\"highlight_red%28+%28x%2B13%2Ai%29%2Ax+%29-%28x%2B13%2Ai%29%2A13%2Ai\"$ .
Expanded term $\"x\"$ by using associative property on $\"%28x%2B13%2Ai%29\"$
It becomes $\"highlight_green%28+x%2Ax+%29%2Bhighlight_green%28+x%2A13%2Ai+%29-%28x%2B13%2Ai%29%2A13%2Ai\"$ .
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Look at $\"highlight_red%28+x+%29%2Ahighlight_red%28+x+%29%2Bx%2A13%2Ai-%28x%2B13%2Ai%29%2A13%2Ai\"$ .
Reduce similar several occurrences of $\"highlight_red%28+x+%29\"$ to $\"highlight_green%28+x%5E2+%29\"$
It becomes $\"highlight_green%28+x%5E2+%29%2Bx%2A13%2Ai-%28x%2B13%2Ai%29%2A13%2Ai\"$ .
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Look at $\"x%5E2%2Bx%2A13%2Ai-highlight_red%28+%28x%2B13%2Ai%29%2A13%2Ai+%29\"$ .
Expanded term $\"-13%2Ai\"$ by using associative property on $\"%28x%2B13%2Ai%29\"$
It becomes $\"x%5E2%2Bx%2A13%2Ai-highlight_green%28+13%2Ai%2Ax+%29-highlight_green%28+13%2Ai%2A13%2Ai+%29\"$ .
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Look at $\"x%5E2%2Bx%2A13%2Ai-13%2Ai%2Ax-highlight_red%28+13+%29%2Ai%2Ahighlight_red%28+13+%29%2Ai\"$ .
Multiplied numerator integers
It becomes $\"x%5E2%2Bx%2A13%2Ai-13%2Ai%2Ax-highlight_green%28+169+%29%2Ai%2Ai\"$ .
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Look at $\"x%5E2%2Bx%2A13%2Ai-13%2Ai%2Ax-169%2Ahighlight_red%28+i+%29%2Ahighlight_red%28+i+%29\"$ .
Reduce similar several occurrences of $\"highlight_red%28+i+%29\"$ to $\"highlight_green%28+i%5E2+%29\"$
It becomes $\"x%5E2%2Bx%2A13%2Ai-13%2Ai%2Ax-169%2Ahighlight_green%28+i%5E2+%29\"$ .
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Look at $\"x%5E2%2Bhighlight_red%28+x%2A13%2Ai+%29-highlight_red%28+13%2Ai%2Ax+%29-169%2Ai%5E2\"$ .
Eliminated similar terms $\"highlight_red%28+x%2A13%2Ai+%29\"$ , $\"highlight_red%28+-13%2Ai%2Ax+%29\"$ replacing them with $\"highlight_green%28+%2813-13%29%2Ax%2Ai+%29\"$
It becomes $\"x%5E2%2Bhighlight_green%28+%2813-13%29%2Ax%2Ai+%29-169%2Ai%5E2\"$ .
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Look at $\"x%5E2%2Bhighlight_red%28+%2813-13%29%2Ax%2Ai+%29-169%2Ai%5E2\"$ .
Since $\"highlight_red%28+%2813-13%29%2Ax%2Ai+%29\"$ has zero as a factor, it should be replaced with a zero
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Look at $\"x%5E2%2B0-169%2Ai%5E2\"$ .
Added fractions or integers together
It becomes $\"x%5E2%2B0-169%2Ai%5E2\"$ .
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Look at $\"x%5E2%2Bhighlight_red%28+0+%29-169%2Ai%5E2\"$ .
Remove extraneous zero $\"highlight_red%28+0+%29\"$
It becomes $\"x%5E2-169%2Ai%5E2\"$ .
Result: $\"x%5E2-169%2Ai%5E2\"$
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Universal Simplifier and Solver
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Done!
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Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:

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\n" ); document.write( " YOUR ANSWER\n" ); document.write( "

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Graphical form: $\"%28x-5%2B10i%29%2A%28x-5-10i%29\"$ simplifies to $\"x%5E2-10%2Ax-100%2Ai%5E2%2B25\"$
Text form: (x-5+10i)*(x-5-10i) simplifies to x^2-10*x-100*i^2+25
Cartoon (animation) form: $\"simplify_cartoon%28+%28x-5%2B10i%29%2A%28x-5-10i%29+%29\"$
For tutors: simplify_cartoon( (x-5+10i)*(x-5-10i) )
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If you have a website, here's a link to this solution.
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\n" ); document.write( "
\n" ); document.write( "
DETAILED EXPLANATION
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Look at $\"%28x-highlight_red%28+5+%29%2B10%2Ai%29%2A%28x-5-10%2Ai%29\"$ .
Moved $\"-5\"$ to the right of expression
It becomes $\"%28x%2B10%2Ai-highlight_green%28+5+%29%29%2A%28x-5-10%2Ai%29\"$ .
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Look at $\"%28x%2B10%2Ai-5%29%2A%28x-highlight_red%28+5+%29-10%2Ai%29\"$ .
Moved $\"-5\"$ to the right of expression
It becomes $\"%28x%2B10%2Ai-5%29%2A%28x-10%2Ai-highlight_green%28+5+%29%29\"$ .
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Look at $\"highlight_red%28+%28x%2B10%2Ai-5%29%2A%28x-10%2Ai-5%29+%29\"$ .
Expanded term $\"%28x-10%2Ai-5%29\"$ by using associative property on $\"%28x%2B10%2Ai-5%29\"$
It becomes .
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Look at $\"highlight_red%28+%28x-10%2Ai-5%29%2Ax+%29%2B%28x-10%2Ai-5%29%2A10%2Ai-%28x-10%2Ai-5%29%2A5\"$ .
Expanded term $\"x\"$ by using associative property on $\"%28x-10%2Ai-5%29\"$
It becomes .
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Look at .
Reduce similar several occurrences of $\"highlight_red%28+x+%29\"$ to $\"highlight_green%28+x%5E2+%29\"$
It becomes .
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Look at $\"x%5E2-x%2A10%2Ai-x%2A5%2Bhighlight_red%28+%28x-10%2Ai-5%29%2A10%2Ai+%29-%28x-10%2Ai-5%29%2A5\"$ .
Expanded term $\"10%2Ai\"$ by using associative property on $\"%28x-10%2Ai-5%29\"$
It becomes .
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Look at .
Multiplied numerator integers
It becomes .
\n" ); document.write( "
Look at .
Multiplied numerator integers
It becomes .
\n" ); document.write( "
Look at .
Reduce similar several occurrences of $\"highlight_red%28+i+%29\"$ to $\"highlight_green%28+i%5E2+%29\"$
It becomes .
\n" ); document.write( "
Look at .
Eliminated similar terms $\"highlight_red%28+-x%2A10%2Ai+%29\"$ , $\"highlight_red%28+10%2Ai%2Ax+%29\"$ replacing them with $\"highlight_green%28+%28-10%2B10%29%2Ax%2Ai+%29\"$
It becomes .
\n" ); document.write( "
Look at .
Since $\"highlight_red%28+%28-10%2B10%29%2Ax%2Ai+%29\"$ has zero as a factor, it should be replaced with a zero
\n" ); document.write( "
Look at $\"x%5E2%2B0-x%2A5-100%2Ai%5E2-50%2Ai-%28x-10%2Ai-5%29%2A5\"$ .
Added fractions or integers together
It becomes $\"x%5E2%2B0-x%2A5-100%2Ai%5E2-50%2Ai-%28x-10%2Ai-5%29%2A5\"$ .
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Look at $\"x%5E2%2Bhighlight_red%28+0+%29-x%2A5-100%2Ai%5E2-50%2Ai-%28x-10%2Ai-5%29%2A5\"$ .
Remove extraneous zero $\"highlight_red%28+0+%29\"$
It becomes $\"x%5E2-x%2A5-100%2Ai%5E2-50%2Ai-%28x-10%2Ai-5%29%2A5\"$ .
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Look at $\"x%5E2-x%2A5-100%2Ai%5E2-50%2Ai-highlight_red%28+%28x-10%2Ai-5%29%2A5+%29\"$ .
Expanded term $\"-5\"$ by using associative property on $\"%28x-10%2Ai-5%29\"$
It becomes .
\n" ); document.write( "
Look at .
Multiplied numerator integers
It becomes $\"x%5E2-x%2A5-100%2Ai%5E2-50%2Ai-5%2Ax%2Bhighlight_green%28+50+%29%2Ai%2B5%2A5\"$ .
\n" ); document.write( "
Look at .
Multiplied numerator integers
It becomes $\"x%5E2-x%2A5-100%2Ai%5E2-50%2Ai-5%2Ax%2B50%2Ai%2Bhighlight_green%28+25+%29\"$ .
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Look at $\"x%5E2-highlight_red%28+x%2A5+%29-100%2Ai%5E2-50%2Ai-highlight_red%28+5%2Ax+%29%2B50%2Ai%2B25\"$ .
Eliminated similar terms $\"highlight_red%28+-x%2A5+%29\"$ , $\"highlight_red%28+-5%2Ax+%29\"$ replacing them with $\"highlight_green%28+%28-5-5%29%2Ax+%29\"$
It becomes $\"x%5E2%2Bhighlight_green%28+%28-5-5%29%2Ax+%29-100%2Ai%5E2-50%2Ai%2B50%2Ai%2B25\"$ .
\n" ); document.write( "
Look at .
Eliminated similar terms $\"highlight_red%28+-50%2Ai+%29\"$ , $\"highlight_red%28+50%2Ai+%29\"$ replacing them with $\"highlight_green%28+%28-50%2B50%29%2Ai+%29\"$
It becomes $\"x%5E2%2B%28-5-5%29%2Ax-100%2Ai%5E2%2Bhighlight_green%28+%28-50%2B50%29%2Ai+%29%2B25\"$ .
\n" ); document.write( "
Look at .
Added fractions or integers together
It becomes $\"x%5E2%2B%28highlight_green%28+-10+%29%29%2Ax-100%2Ai%5E2%2B%28-50%2B50%29%2Ai%2B25\"$ .
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Look at $\"x%5E2%2B%28-10%29%2Ax-100%2Ai%5E2%2Bhighlight_red%28+%28-50%2B50%29%2Ai+%29%2B25\"$ .
Since $\"highlight_red%28+%28-50%2B50%29%2Ai+%29\"$ has zero as a factor, it should be replaced with a zero
\n" ); document.write( "
Look at $\"x%5E2%2B%28-10%29%2Ax-100%2Ai%5E2%2B0%2B25\"$ .
Added fractions or integers together
It becomes $\"x%5E2%2B%28-10%29%2Ax-100%2Ai%5E2%2B0%2B25\"$ .
\n" ); document.write( "
Look at $\"x%5E2%2B%28-10%29%2Ax-100%2Ai%5E2%2Bhighlight_red%28+0+%29%2B25\"$ .
Remove extraneous zero $\"highlight_red%28+0+%29\"$
It becomes $\"x%5E2%2B%28-10%29%2Ax-100%2Ai%5E2%2B25\"$ .
\n" ); document.write( "
Look at $\"x%5E2%2B%28highlight_red%28+-10+%29%29%2Ax-100%2Ai%5E2%2B25\"$ .
Removed extra sign in front of $\"-10\"$
It becomes $\"x%5E2%2B%28-highlight_green%28+10+%29%29%2Ax-100%2Ai%5E2%2B25\"$ .
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Look at $\"x%5E2%2Bhighlight_red%28+%28-highlight_red%28+10+%29%29%2Ax+%29-100%2Ai%5E2%2B25\"$ .
Remove unneeded parentheses around factor $\"highlight_red%28+10+%29\"$
It becomes $\"x%5E2-highlight_green%28+10+%29%2Ax-100%2Ai%5E2%2B25\"$ .
Result: $\"x%5E2-10%2Ax-100%2Ai%5E2%2B25\"$
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Universal Simplifier and Solver
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Done!

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Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:

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\n" ); document.write( " YOUR ANSWER\n" ); document.write( "

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Graphical form: $\"%28x%5E2%2B169%29%2A%28x%5E2-10x%2B125%29\"$ simplifies to $\"x%5E%284%29-x%5E%283%29%2A10%2B294%2Ax%5E2-1690%2Ax%2B21125\"$
Text form: (x^2+169)*(x^2-10x+125) simplifies to x^(4)-x^(3)*10+294*x^2-1690*x+21125
Cartoon (animation) form: $\"simplify_cartoon%28+%28x%5E2%2B169%29%2A%28x%5E2-10x%2B125%29+%29\"$
For tutors: simplify_cartoon( (x^2+169)*(x^2-10x+125) )
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If you have a website, here's a link to this solution.
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\n" ); document.write( "
DETAILED EXPLANATION
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Look at $\"highlight_red%28+%28x%5E2%2B169%29%2A%28x%5E2-10%2Ax%2B125%29+%29\"$ .
Expanded term $\"%28x%5E2-10%2Ax%2B125%29\"$ by using associative property on $\"%28x%5E2%2B169%29\"$
It becomes .
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Look at $\"highlight_red%28+%28x%5E2-10%2Ax%2B125%29%2Ax%5E2+%29%2B%28x%5E2-10%2Ax%2B125%29%2A169\"$ .
Expanded term $\"x%5E2\"$ by using associative property on $\"%28x%5E2-10%2Ax%2B125%29\"$
It becomes .
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Look at .
Reduce similar several occurrences of $\"highlight_red%28+x%5E2+%29\"$ to $\"highlight_green%28+x%5E%282%2B2%29+%29\"$
It becomes .
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Look at .
Reduce similar several occurrences of $\"highlight_red%28+x%5E2+%29\"$ to $\"highlight_green%28+x%5E%281%2B2%29+%29\"$
It becomes .
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Look at .
Added fractions or integers together
It becomes .
\n" ); document.write( "
Look at .
Added fractions or integers together
It becomes .
\n" ); document.write( "
Look at .
Expanded term $\"169\"$ by using associative property on $\"%28x%5E2-10%2Ax%2B125%29\"$
It becomes .
\n" ); document.write( "
Look at .
Multiplied numerator integers
It becomes .
\n" ); document.write( "
Look at .
Multiplied numerator integers
It becomes .
\n" ); document.write( "
Look at .
Eliminated similar terms $\"highlight_red%28+x%5E2%2A125+%29\"$ , $\"highlight_red%28+169%2Ax%5E2+%29\"$ replacing them with $\"highlight_green%28+%28125%2B169%29%2Ax%5E2+%29\"$
It becomes .
\n" ); document.write( "
Look at .
Added fractions or integers together
It becomes $\"x%5E%284%29-x%5E%283%29%2A10%2B%28highlight_green%28+294+%29%29%2Ax%5E2-1690%2Ax%2B21125\"$ .
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Look at .
Remove unneeded parentheses around factor $\"highlight_red%28+294+%29\"$
It becomes $\"x%5E%284%29-x%5E%283%29%2A10%2Bhighlight_green%28+294+%29%2Ax%5E2-1690%2Ax%2B21125\"$ .
Result: $\"x%5E%284%29-x%5E%283%29%2A10%2B294%2Ax%5E2-1690%2Ax%2B21125\"$
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Universal Simplifier and Solver
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Done!
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