Final Review U5 Polynomials

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. It can be expressed in the form: $$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n, a_{n-1}, ..., a_0$$ are constants.

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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. For example, in the polynomial $$5a^4 + 7a^2 - 3a^9 + 10$$, the degree is 9.

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What is a leading term in a polynomial?

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The leading term of a polynomial is the term with the highest degree. For example, in the polynomial $$5a^4 + 7a^2 - 3a^9 + 10$$, the leading term is $$-3a^9$$.

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What are the zeros of a polynomial?

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The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. For example, for the polynomial $$f(x) = x(x+3)^2(x-2)$$, the zeros are 0, -3 (with multiplicity 2), and 2.

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What is multiplicity in the context of polynomial zeros?

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Multiplicity refers to the number of times a particular zero occurs in a polynomial. For example, if a zero occurs twice, it has a multiplicity of 2.

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How do you determine if a leading coefficient is positive or negative?

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The leading coefficient is the coefficient of the term with the highest degree. If it is greater than zero, the leading coefficient is positive; if it is less than zero, it is negative.

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What is an increasing interval of a polynomial function?

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An increasing interval is a range of x-values where the function's output (y-value) is rising as x increases. For example, if a function is increasing on the interval (2, 5), then as x moves from 2 to 5, the function's value increases.

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What is a decreasing interval of a polynomial function?

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A decreasing interval is a range of x-values where the function's output (y-value) is falling as x increases. For example, if a function is decreasing on the interval (0, 2), then as x moves from 0 to 2, the function's value decreases.

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What is the standard form of a polynomial?

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The standard form of a polynomial is when it is written in descending order of the degree of its terms. For example, $$3x^4 + 2x^3 - x + 5$$ is in standard form.

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What is the factored form of a polynomial?

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