PC Ch 2 - Zeros of Polynomial Functions

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What is a zero of a polynomial function?

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A zero of a polynomial function is a value of x for which the polynomial evaluates to zero, meaning P(x) = 0.

Tags

CCSS.HSF-IF.C.7C

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, you can use methods such as factoring, synthetic division, or applying the Rational Root Theorem.

Tags

CCSS.HSF-IF.C.7C

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c).

Tags

CCSS.HSA.APR.D.6

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root, and the total number of roots (including multiplicities) is equal to the degree of the polynomial.

Tags

CCSS.HSN.CN.C.9

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What are complex roots?

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Complex roots are solutions to polynomial equations that include imaginary numbers, typically in the form a + bi, where a and b are real numbers.

Tags

CCSS.HSN.CN.C.9

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If a polynomial has real coefficients, what can be said about its complex roots?

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If a polynomial has real coefficients, then any complex roots must occur in conjugate pairs, meaning if a + bi is a root, then a - bi is also a root.

Tags

CCSS.HSN.CN.C.9

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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 expression.

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How do you determine the number of possible real zeros of a polynomial?

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The number of possible real zeros of a polynomial can be determined by its degree; a polynomial of degree n can have up to n real zeros.

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

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The zeros of a polynomial correspond to the values of x that make each factor equal to zero, meaning if (x - r) is a factor, then r is a zero.

Tags

CCSS.HSF-IF.C.7C

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

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The leading coefficient of a polynomial affects the end behavior of the graph; it determines whether the graph rises or falls as x approaches positive or negative infinity.