5.2 Writing Polynomial Equations Given Roots

Front

What is a polynomial equation?

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A polynomial equation is an equation that involves a polynomial expression, which is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient.

Front

What are the roots of a polynomial?

Back

The roots of a polynomial are the values of the variable that make the polynomial equal to zero.

Front

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 the roots are r1, r2, ..., rn, the polynomial can be expressed as (x - r1)(x - r2)...(x - rn).

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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 real coefficients, any complex roots must appear as pairs of the form a + bi and a - bi.

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What is the standard form of a polynomial?

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The standard form of a polynomial is expressed as a sum of terms in descending order of the degree of the variable, typically written as a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0.

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What does it mean for a root to have multiplicity?

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Multiplicity refers to the number of times a particular root appears in a polynomial. A root with multiplicity greater than 1 indicates that the root is repeated.

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How do you write a polynomial from given roots with multiplicity?

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If a root r has multiplicity m, it is included in the polynomial as (x - r)^m. For example, if r = 3 with multiplicity 2, it contributes (x - 3)^2 to the polynomial.

Front

What is the relationship between the degree of a polynomial and its roots?

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The degree of a polynomial is equal to the number of roots it has, counting multiplicities. For example, a polynomial of degree 3 has 3 roots.

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How do you expand a polynomial from its factored form?

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To expand a polynomial from its factored form, use the distributive property (FOIL for binomials) to multiply the factors together until you reach a standard polynomial form.

Front

What is the quadratic formula?

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The quadratic formula is used to find the roots of a quadratic equation ax^2 + bx + c = 0, given by x = (-b ± √(b² - 4ac)) / (2a).