A2-4.7 FLASHCARD 2023-2024 Writing Polynomials given Zeros/Roots

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. It can be expressed in the form: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants.

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What does it mean for a polynomial to be in standard form?

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A polynomial is in standard form when it is written as a sum of terms in descending order of the degree of the variable. For example, f(x) = 3x^4 + 2x^3 - x + 5 is in standard form.

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How do you find a polynomial given its zeros?

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To find a polynomial given its zeros, use the fact that if r is a zero, then (x - r) is a factor of the polynomial. Multiply the factors corresponding to each zero to obtain the polynomial.

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What is the least degree of a polynomial?

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The least degree of a polynomial is the smallest degree of its terms when expressed in standard form. It corresponds to the number of distinct zeros the polynomial has.

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What are integral coefficients in a polynomial?

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Integral coefficients are the numerical factors of the terms in a polynomial that are whole numbers (positive, negative, or zero). For example, in the polynomial 2x^3 - 3x + 5, the coefficients are 2, -3, and 5.

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Write the polynomial in standard form given the zeros x = -2, 1, 4.

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f(x) = x^3 - 3x^2 - 6x + 8.

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What is the relationship between zeros and factors of a polynomial?

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The zeros of a polynomial are the values of x that make the polynomial equal to zero. Each zero corresponds to a factor of the form (x - r), where r is the zero.

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How do you determine the degree of a polynomial?

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The degree of a polynomial is determined by the highest power of the variable in the polynomial. For example, in f(x) = 4x^5 + 3x^2 - 2, the degree is 5.

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What is the significance of the leading coefficient in a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial's graph.

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How can you verify if a polynomial has the correct zeros?

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To verify if a polynomial has the correct zeros, substitute each zero into the polynomial. If the polynomial evaluates to zero for each zero, then they are correct.