4.5A Practice - Solving Polynomial Equations

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Define 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 are the zeros of a polynomial function?

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The zeros of a polynomial function are the values of $$x$$ for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the polynomial equal to zero and solve for $$x$$. This can involve factoring, using the quadratic formula, or applying synthetic division.

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What is the relationship between the degree of a polynomial and the number of zeros?

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The degree of a polynomial indicates the maximum number of zeros (roots) it can have. A polynomial of degree $$n$$ can have up to $$n$$ real or complex zeros.

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Explain the concept of complex zeros in polynomials.

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Complex zeros occur in conjugate pairs for polynomials with real coefficients. If $$a + bi$$ is a zero, then $$a - bi$$ is also a zero, where $$i$$ is the imaginary unit.

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Write a polynomial function with zeros -2, 5, -3i, and 3i.

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The polynomial function is: $$f(x) = x^4 - 3x^3 - x^2 - 27x - 90$$.

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What are the roots of the polynomial $$f(x) = (2x-1)(x+7)(x-3)$$?

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The roots are: $$-7, \frac{1}{2}, 3$$.

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How do you determine the roots from a factored polynomial?

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To determine the roots from a factored polynomial, set each factor equal to zero and solve for $$x$$.

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What is the significance of integral coefficients in polynomial functions?

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Integral coefficients mean that all the coefficients in the polynomial are whole numbers (positive, negative, or zero). This ensures that the polynomial can be expressed without fractions.

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Write a polynomial function with zeros 4, -4, i, and -i.

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The polynomial function is: $$f(x) = x^4 - 15x^2 - 16$$.