5.2 Writing Polynomial Equations Given 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 whole number powers and multiplied by coefficients. For example, 2x^2 + 3x + 1 is a polynomial.

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What are the roots of a polynomial?

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The roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as zeros or solutions of the polynomial.

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How do you find a polynomial given its roots?

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To find a polynomial given its roots, you can use the factored form: if r1, r2, ..., rn are the roots, the polynomial can be expressed as (x - r1)(x - r2)...(x - rn).

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 3x^4 + 2x^3 - x + 5, the degree is 4.

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What is the standard form of a polynomial?

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The standard form of a polynomial is when it is written in descending order of the degree of its terms. For example, 2x^3 + 3x^2 - x + 4 is in standard form.

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its factors. For example, x^2 - 5 can be factored as (x - √5)(x + √5).

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

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If r is a root of a polynomial, then (x - r) is a factor of that polynomial. This means that if you know the roots, you can write the polynomial in factored form.

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How do you write a polynomial with given roots?

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To write a polynomial with given roots, use the roots to create factors of the form (x - root). Multiply these factors together to get the polynomial.

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What is the significance of complex roots in polynomials?

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Complex roots occur in conjugate pairs. If a polynomial has a complex root a + bi, it must also have the root a - bi. This is important for ensuring that polynomials with real coefficients have real solutions.

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What is the polynomial function of least degree?

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The polynomial function of least degree with given roots is the simplest polynomial that has those roots, typically found by multiplying the factors corresponding to the roots.