Writing Polynomials given Zeros/Roots

Front

What is a polynomial?

Back

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?

Back

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

Front

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

Back

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

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.

Front

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.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression.

Front

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.

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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.

Front

What is the significance of integral coefficients in polynomials?

Back

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.