What does it mean if a root has an even multiplicity?

If a root has an even multiplicity, the graph of the polynomial touches the x-axis at that root but does not cross it. For example, if $$x = 2$$ is a root with multiplicity 2, the graph will touch the x-axis at $$x = 2$$ .

What does it mean if a root has an odd multiplicity?

If a root has an odd multiplicity, the graph of the polynomial crosses the x-axis at that root. For example, if $$x = 3$$ is a root with multiplicity 3, the graph will cross the x-axis at $$x = 3$$ .

How do you express a polynomial given its roots and their multiplicities?

To express a polynomial given its roots and their multiplicities, use the form $$f(x) = k(x - r_1)^{m_1}(x - r_2)^{m_2}...$$ where $$r_i$$ are the roots and $$m_i$$ are their respective multiplicities.

What is the polynomial function for roots -2 and 7 with multiplicity 3?

The polynomial function is $$f(x) = (x + 2)(x - 7)^3$$ .

How do you determine the number of real roots of a polynomial?

The number of real roots of a polynomial can be determined by analyzing its degree and the behavior of the polynomial graph. A polynomial of degree n can have up to n real roots.

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

The leading coefficient of a polynomial (the coefficient of the term with the highest degree) affects the end behavior of the polynomial's graph. If positive, the graph rises to the right; if negative, it falls to the right.

What is the factored form of a polynomial?

The factored form of a polynomial expresses it as a product of its factors. For example, $$f(x) = (x - 1)(x + 2)(x - 3)$$ is a factored form.

How do you find the zeros of a polynomial from its factored form?

To find the zeros of a polynomial from its factored form, set each factor equal to zero and solve for the variable. For example, from $$(x - 1)(x + 2) = 0$$ , the zeros are $$x = 1$$ and $$x = -2$$ .

Zeros, Multiplicities, and Degrees of Polynomials Flashcards

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FLASHCARD QUESTION

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What is a polynomial?

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2.

FLASHCARD QUESTION

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What are the roots of a polynomial?

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