Polynomial Functions Review

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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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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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. For example, in the polynomial $$2x^3 - 4x^2 + 5$$, the degree is 3.

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What is the domain of a polynomial function?

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The domain of a polynomial function is all real numbers, as polynomials are defined for every real value of the variable.

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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 equals 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 synthetic division.

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What is end behavior of polynomial functions?

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The end behavior of a polynomial function describes how the function behaves as $$x$$ approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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What is the leading coefficient of a polynomial?

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The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It influences the end behavior of the polynomial.

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form $$x - c$$. It is faster than long division and is used to find polynomial zeros.

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Simplify: $$\left(3k^2-8k-5\right)\left(6k-1\right)$$

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$$18k^3 - 51k^2 - 22k + 5$$

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Simplify: $$\left(2x^2y^4\right)^3$$

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$$8x^6y^{12}$$