Writing Polynomials given Zeros/Roots

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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What does it mean for a polynomial to have integral coefficients?

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Integral coefficients mean that the coefficients (the numbers in front of the variables) of the polynomial are whole numbers (positive, negative, or zero).

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CCSS.HSA.APR.D.6

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

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The zeros (or roots) of a polynomial are the values of x that make the polynomial equal to zero. Each zero corresponds to a factor of the polynomial in the form (x - zero).

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CCSS.HSF-IF.C.7C

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If a polynomial has a complex zero, what can be said about its other zeros?

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If a polynomial has a complex zero, its conjugate must also be a zero. For example, if 2 + 3i is a zero, then 2 - 3i is also a zero.

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CCSS.HSN.CN.C.9

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How do you find a polynomial given its zeros?

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To find a polynomial given its zeros, write the polynomial as a product of factors corresponding to each zero. For example, if the zeros are r1, r2, and r3, the polynomial can be expressed as f(x) = k(x - r1)(x - r2)(x - r3), where k is a constant.

Tags

CCSS.HSF-IF.C.7C

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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, f(x) = 2x^3 - 4x^2 + 3x - 5.

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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 4x^3 - 2x + 7, the degree is 3.

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

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The leading coefficient is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial graph.

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How do you determine the number of zeros a polynomial can have?

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A polynomial of degree n can have at most n zeros, counting multiplicities. This means if a zero is repeated, it is counted multiple times.

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What is a root of a polynomial?

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A root of a polynomial is a value of x that makes the polynomial equal to zero. It is also referred to as a zero.

Tags

CCSS.HSF-IF.C.7C