How to form a polynomial with given zeros and degree?
Understand the Problem
The question is asking how to construct a polynomial expression given specific zeros (roots) and the degree of the polynomial. This typically involves using the fact that if a polynomial has certain zeros, it can be expressed in a factored form that incorporates those zeros. The degree of the polynomial indicates how many zeros (or roots) it has.
Answer
The polynomial can be expressed as $$ P(x) = k(x - r_1)(x - r_2)...(x - r_n $$
Answer for screen readers
Given the zeros ( r_1, r_2, \ldots, r_n ) and a degree ( n ), the polynomial can be written as: $$ P(x) = k(x - r_1)(x - r_2)...(x - r_n $$
Steps to Solve
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Identifying the Zeros The first step is to note down the zeros (roots) of the polynomial. Let's say the zeros are ( r_1, r_2, ..., r_n ).
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Construct the Factored Form Using the identified zeros, we can construct the polynomial in factored form. For each zero ( r_i ), we include a factor of ( (x - r_i) ). If we have ( n ) zeros, the polynomial can be expressed as: $$ P(x) = k(x - r_1)(x - r_2)...(x - r_n $$ where ( k ) is a constant, often set to 1 if not specified.
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Determining the Degree The degree of the polynomial is equal to the number of its zeros. Make sure that the number of roots matches the given degree of the polynomial.
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Expanding the Factored Form If required, expand the factored form to get the polynomial in standard form, which means combining like terms.
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Final Check Make sure that the polynomial matches the specified degree and has the correct zeros.
Given the zeros ( r_1, r_2, \ldots, r_n ) and a degree ( n ), the polynomial can be written as: $$ P(x) = k(x - r_1)(x - r_2)...(x - r_n $$
More Information
The polynomial constructed using its zeros is rooted in the Fundamental Theorem of Algebra, which states that a polynomial of degree ( n ) will have exactly ( n ) roots (considering multiplicities). This is useful in various areas of mathematics, including calculus and numerical methods.
Tips
- Forgetting to include all zeros: Ensure that you have accounted for all zeros, especially if the degree is higher than the number of provided unique zeros.
- Not properly expanding the factors: When expanding the factored form, group like terms carefully to avoid mistakes in coefficients.
