Writing Polynomials given Zeros/Roots

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

Front

What does it mean for a polynomial to have zeros/roots?

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Zeros or roots of a polynomial are the values of the variable that make the polynomial equal to zero.

Tags

CCSS.HSF-IF.C.7C

Front

If a polynomial has a zero at x = r, what can be said about the factor of the polynomial?

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If x = r is a zero of the polynomial, then (x - r) is a factor of the polynomial.

Tags

CCSS.HSA.APR.B.2

Front

What is the relationship between real and imaginary roots of polynomials?

Back

Imaginary roots of polynomials always come in conjugate pairs. If a + bi is a root, then a - bi is also a root.

Tags

CCSS.HSN.CN.C.9

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How do you write a polynomial given its zeros?

Back

To write a polynomial given its zeros, multiply the factors corresponding to each zero. For example, if the zeros are r1, r2, and r3, the polynomial is P(x) = k(x - r1)(x - r2)(x - r3), where k is a constant.

Tags

CCSS.HSA.APR.B.3

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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 expression.

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What is the least degree polynomial with given zeros?

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The least degree polynomial with given zeros is the polynomial formed by multiplying the factors corresponding to each zero, without any repeated factors.

Tags

CCSS.HSF-IF.C.7C

Front

How do you find a polynomial function of degree 3 with specific zeros?

Back

To find a polynomial function of degree 3 with specific zeros, multiply the factors corresponding to the zeros, ensuring to include complex conjugates if necessary.

Tags

CCSS.HSF-IF.C.7C

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What is the significance of integral coefficients in polynomials?

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Integral coefficients mean that the coefficients of the polynomial are whole numbers (positive, negative, or zero), which can affect the nature of the roots.

Front

What is the conjugate root theorem?

Back

The conjugate root theorem states that if a polynomial has real coefficients and a complex root a + bi, then its conjugate a - bi is also a root.

Tags

CCSS.HSN.CN.C.9