Roots of Polynomials

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What are the zeros (roots, x-intercepts) of a polynomial function?

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The zeros of a polynomial function are the values of x for which the function equals zero. They can be found by solving the equation f(x) = 0.

Tags

CCSS.HSF-IF.C.7C

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To find the roots, factor the equation: (x + 6)(x - 3) = 0. The roots are x = 3 and x = -6.

Tags

CCSS.HSA-REI.B.4B

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What is a relative maximum in the context of polynomial functions?

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A relative maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.

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What are the roots of the polynomial represented by the factors (x-4)(x+7)?

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The roots are x = 4 and x = -7.

Tags

CCSS.HSF-IF.C.7C

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Factor out an x: x(x^2 - 4) = 0. The roots are x = 0, x = 2, and x = -2.

Tags

CCSS.HSF-IF.C.7C

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What is the relationship between the degree of a polynomial and the number of roots it can have?

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A polynomial of degree n can have at most n roots, counting multiplicities.

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What is the significance of the x-intercepts of a polynomial function?

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The x-intercepts (or roots) indicate where the graph of the function crosses the x-axis, representing the values of x for which the function equals zero.

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How can you determine the number of real roots of a quadratic equation?

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Use the discriminant (b^2 - 4ac) from the quadratic formula. If the discriminant is positive, there are two real roots; if zero, one real root; if negative, no real roots.

Tags

CCSS.HSA-REI.B.4B

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What is the quadratic formula used for?

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Tags

CCSS.HSA-REI.B.4B

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What does it mean for a polynomial to be factored completely?

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A polynomial is factored completely when it is expressed as a product of irreducible polynomials over the real numbers.