Use Rational Root Theorem to find the Zeros of the Polynomial

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

Tags

CCSS.HSA.APR.B.2

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How do you determine if a polynomial has a rational root?

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To determine if a polynomial has a rational root, use the Rational Root Theorem to list all possible rational roots, then test these roots using synthetic division or substitution.

Tags

CCSS.HSA.APR.B.2

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form (x - c). It is quicker and requires less writing.

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CCSS.HSA.APR.D.6

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What does it mean if the remainder is zero in synthetic division?

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If the remainder is zero when using synthetic division, it means that the divisor (x - c) is a factor of the polynomial, and c is a root of the polynomial.

Tags

CCSS.HSA.APR.D.6

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Given f(x) = 2x^3 - 19x^2 + 38x + 24, if (x - 4) is a factor, what is one zero of the function?

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One zero of the function is x = 4.

Tags

CCSS.HSA.APR.B.2

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How can you find all zeros of a polynomial?

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To find all zeros of a polynomial, first identify any rational roots using the Rational Root Theorem, then use synthetic division to factor the polynomial, and finally solve the resulting factors.

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What are the possible rational zeros of a polynomial?

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The possible rational zeros of a polynomial can be found by taking the factors of the constant term and dividing them by the factors of the leading coefficient.

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If P(x) = x^3 - 3x^2 - 10x + 24, what are the zeros?

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The zeros of P(x) are -3, 2, and 4.

Tags

CCSS.HSF-IF.C.7C

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What is the significance of finding zeros of a polynomial?

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Finding the zeros of a polynomial helps in understanding the behavior of the function, including its intercepts and turning points.

Tags

CCSS.HSF-IF.C.7C

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What is a factor of a polynomial?

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A factor of a polynomial is a polynomial of lower degree that divides the original polynomial without leaving a remainder.

Tags

CCSS.HSA.APR.B.2