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 that makes the polynomial equal to zero.

Tags

CCSS.HSF-IF.C.7C

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How can you find rational zeros of a polynomial?

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Use the Rational Root Theorem, which states that any rational solution, expressed as a fraction p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

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What is synthetic division?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor.

Tags

CCSS.HSA.APR.D.6

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What does it mean if a polynomial has a repeated root?

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A repeated root means that the root is a solution to the polynomial equation more than once, indicating that the factor corresponding to that root appears multiple times in the polynomial.

Tags

CCSS.HSF-IF.C.7C

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How do you verify if a given polynomial factor is correct?

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Multiply the factors together to see if you obtain the original polynomial.

Tags

CCSS.HSA.APR.D.6

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What is the significance of the remainder in synthetic division?

Back

If the remainder is zero, it indicates that the divisor is a factor of the polynomial.

Tags

CCSS.HSA.APR.D.6

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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.

Tags

CCSS.HSA.APR.B.2

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How do you factor a polynomial when one root is known?

Back

Use synthetic division to divide the polynomial by (x - known root) to find the quotient, which can then be factored further.

Tags

CCSS.HSA.APR.B.2

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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 at most n zeros, counting multiplicities.

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What is the purpose of the Rational Root Theorem?

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It helps to identify possible rational zeros of a polynomial, which can then be tested to find actual zeros.