How to find a polynomial with given zeros?
Understand the Problem
The question is asking how to construct a polynomial equation when the zeros (or roots) are provided. This typically involves creating factors from the zeros and multiplying them to form the polynomial.
Answer
The polynomial from the roots \( r_1, r_2, r_3 \) is given by $$ P(x) = x^3 - (r_1+r_2+r_3)x^2 + (r_1r_2 + r_1r_3 + r_2r_3)x - r_1r_2r_3 $$
Answer for screen readers
The polynomial ( P(x) ) formed from the roots ( r_1, r_2, r_3 ) is: $$ P(x) = x^3 - (r_1+r_2+r_3)x^2 + (r_1r_2 + r_1r_3 + r_2r_3)x - r_1r_2r_3 $$
Steps to Solve
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Identify the Zeroes First, we need to identify the given zeroes of the polynomial. For example, let's say the zeroes are ( r_1, r_2, ) and ( r_3 ).
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Construct Factors from Zeroes For each zero ( r ), create a factor of the form ( (x - r) ). If there are three zeroes ( r_1, r_2, ) and ( r_3 ), the factors will be: $$ (x - r_1)(x - r_2)(x - r_3) $$
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Multiply the Factors Now, we need to multiply these factors together to form the polynomial. Start with the first two factors and multiply them, then multiply the result by the third factor. For example: $$ P(x) = (x - r_1)(x - r_2)(x - r_3) $$
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Expand the Expression Distribute the factors carefully. If we multiply ( (x - r_1) ) and ( (x - r_2) ) first: $$ (x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + r_1r_2 $$ Next, multiply this result by ( (x - r_3) ): $$ P(x) = (x^2 - (r_1 + r_2)x + r_1r_2)(x - r_3) $$
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Perform the Final Multiplication Distribute ( (x - r_3) ) through your polynomial: $$ P(x) = x^3 - (r_1 + r_2 + r_3)x^2 + (r_1r_2 + r_1r_3 + r_2r_3)x - r_1r_2r_3 $$
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Write the Final Polynomial The final polynomial will be written in the form: $$ P(x) = x^3 - (sum \ of \ zeros)x^2 + (sum \ of \ product \ of \ zeros \ taken \ two \ at \ a \ time)x - (product \ of \ zeros) $$
The polynomial ( P(x) ) formed from the roots ( r_1, r_2, r_3 ) is: $$ P(x) = x^3 - (r_1+r_2+r_3)x^2 + (r_1r_2 + r_1r_3 + r_2r_3)x - r_1r_2r_3 $$
More Information
This method of constructing a polynomial from its roots is useful in algebra for understanding polynomial functions and their graphs. The resulting polynomial gives us information about the behavior and simplified representation of the values where the polynomial intersects the x-axis.
Tips
- Confusing the signs while creating factors. Remember to use ( (x - r) ) for every zero ( r ).
- Forgetting to expand correctly after multiplying factors. It's important to keep track of all terms during multiplication.
