Finding Zeros of Polynomials

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What is a zero of a polynomial?

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A zero of a polynomial is a value of x for which the polynomial evaluates to zero. It is also known as a root of the polynomial.

Tags

CCSS.HSF-IF.C.7C

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What is the Factor Theorem?

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The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0, meaning c is a zero of the polynomial.

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CCSS.HSA.APR.B.2

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How do you find the zeros of a polynomial using the Factor Theorem?

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To find the zeros using the Factor Theorem, substitute potential zeros into the polynomial. If the result is zero, then that value is a zero of the polynomial.

Tags

CCSS.HSA.APR.B.2

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What is the significance of the multiplicity of a zero?

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The multiplicity of a zero indicates how many times that zero is repeated as a root of the polynomial. A zero with an odd multiplicity will cross the x-axis, while one with an even multiplicity will touch the x-axis.

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What is synthetic division and how is it used in finding zeros?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor. It can help find zeros by simplifying the polynomial and identifying factors.

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CCSS.HSA.APR.D.6

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What is the relationship between the degree of a polynomial and the number of zeros?

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A polynomial of degree n can have up to n zeros, counting multiplicities. This means that a polynomial of degree 4 can have up to 4 zeros.

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What is the Remainder Theorem?

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The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder of that division is f(c). If f(c) = 0, then (x - c) is a factor.

Tags

CCSS.HSA.APR.B.2

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How can you verify if a polynomial has a specific factor?

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To verify if a polynomial has a specific factor, use the Factor Theorem: substitute the value that makes the factor zero into the polynomial. If the result is zero, the factor exists.

Tags

CCSS.HSA.APR.B.2

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What is the process for factoring a polynomial?

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Factoring a polynomial involves expressing it as a product of its factors, which can include linear factors and irreducible quadratic factors.

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What is a polynomial function?

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A polynomial function is a function that 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 and n is a non-negative integer.