SOLUTION: I need help figuring out this problem. What is the polynomial function? 13i and 5-10i

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Question 691232: I need help figuring out this problem. What is the polynomial function? 13i and 5-10i
Answer by fcabanski(1391) About Me  (Show Source):
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.


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.


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.


(x-13i)(x+13i) = x^2 +169. The detailed explanation is after the problem solution. Remember i^2 = -1.


(x-5+10i)(x-5-10i)=x^2 -10x +125


The two factors remaining are those answers (x^2 +169)(x^2 -10x +125)= x^4 -10x^3 +294x^2 -1690x +21125

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.)

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


YOUR ANSWER


  • 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) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

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.
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.
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.
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.
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.
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.
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.
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
Look at x%5E2%2B0-169%2Ai%5E2.
Added fractions or integers together
It becomes x%5E2%2B0-169%2Ai%5E2.
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

Universal Simplifier and Solver


Done!


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


YOUR ANSWER


  • 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) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

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.
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.
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 .
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 .
Look at .
Reduce similar several occurrences of highlight_red%28+x+%29 to highlight_green%28+x%5E2+%29
It becomes .
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 .
Look at .
Multiplied numerator integers
It becomes .
Look at .
Multiplied numerator integers
It becomes .
Look at .
Reduce similar several occurrences of highlight_red%28+i+%29 to highlight_green%28+i%5E2+%29
It becomes .
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 .
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
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.
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.
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 .
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.
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.
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.
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.
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.
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
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.
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.
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.
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

Universal Simplifier and Solver


Done!

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


YOUR ANSWER


  • 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) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

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 .
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 .
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 .
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 .
Look at .
Added fractions or integers together
It becomes .
Look at .
Added fractions or integers together
It becomes .
Look at .
Expanded term 169 by using associative property on %28x%5E2-10%2Ax%2B125%29
It becomes .
Look at .
Multiplied numerator integers
It becomes .
Look at .
Multiplied numerator integers
It becomes .
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 .
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.
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

Universal Simplifier and Solver


Done!