Find the polynomial having roots at $ 1 $ , $ -1 $ , $ 3 $ , $ 3 $ , $ -\dfrac{ 2 }{ 3 } $ , $ 2 $ and $ 4 $ .
Answer
The polynomial generated from the given zeros is: $ P(x) = 3x^7-34x^6+132x^5-166x^4-123x^3+344x^2-12x-144 $
Explanation
Step 1: Turn the zeros into factors:
| Zero | Factor |
| $ 1 $ | $ x-1 $ |
| $ -1 $ | $ x+1 $ |
| $ 3 $ | $ x-3 $ |
| $ 3 $ | $ x-3 $ |
| $ -\dfrac{ 2 }{ 3 } $ | $ 3x+2 $ |
| $ 2 $ | $ x-2 $ |
| $ 4 $ | $ x-4 $ |
Step 2: Multiply the factors together:
$$ P(x) = \left( x-1 \right) \cdot \left( x+1 \right) \cdot \left( x-3 \right) \cdot \left( x-3 \right) \cdot \left( 3x+2 \right) \cdot \left( x-2 \right) \cdot \left( x-4 \right) = 3x^7-34x^6+132x^5-166x^4-123x^3+344x^2-12x-144 $$This page was created using
Polynomial From Roots Calculator