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3.E: Polynomial and Rational Functions(Exercises)
https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/03%3A_Polynomial_and_Rational_Functions/3.E%3A_Polynomial_and_Rational_Functions(Exercises)
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this...After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
3.1: Complex Numbers Review
https://math.libretexts.org/Courses/Highline_College/Math_141%3A_Precalculus_I_(old_edition)/03%3A_Polynomial_and_Rational_Functions/3.01%3A_Complex_Numbers_Review
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this...After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
2.23: Complex Numbers
https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/02%3A_Algebra_Support/2.23%3A_Complex_Numbers
\(\begin{array}{cccc}{i^{5}} \qquad& {i^{6}} \qquad& {i^{7}} \qquad& {i^{8}} \\ {i^{4} \cdot i} \qquad& {i^{4} \cdot i^{2}} \qquad& {i^{4} \cdot i^{3}} \qquad& {i^{4} \cdot i^{4}} \\ {1 \cdot i} \qqua... $\begin{array}{cccc}{i^{5}} \qquad& {i^{6}} \qquad& {i^{7}} \qquad& {i^{8}} \\ {i^{4} \cdot i} \qquad& {i^{4} \cdot i^{2}} \qquad& {i^{4} \cdot i^{3}} \qquad& {i^{4} \cdot i^{4}} \\ {1 \cdot i} \qquad& {1 \cdot i^{2}} \qquad& {1 \cdot i^{3}} \qquad& {1 \cdot 1} \\ {i} \qquad& {i^{2}} & {i^{3}} \qquad& {1} \\ {}&\qquad&{-1} & {-i}\end{array}$
2.6: Complex Zeros
https://math.libretexts.org/Courses/North_Hennepin_Community_College/Math_1120%3A_College_Algebra_(Lang)/02%3A_Polynomial_and_Rational_Functions./2.06%3A_Complex_Zeros
When finding the zeros of polynomials, at some point you’re faced with the problem x²=−1 . While there are clearly no real numbers that are solutions to this equation, leaving things there has a cert...When finding the zeros of polynomials, at some point you’re faced with the problem x²=−1 . While there are clearly no real numbers that are solutions to this equation, leaving things there has a certain feel of incompleteness. To address that, we will need utilize the imaginary unit, i .
3.2: Complex Numbers
https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_2e_(OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.02%3A_Complex_Numbers
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this...After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
3.6: Complex Zeros
https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/03%3A_Polynomial_and_Rational_Functions./3.06%3A_Complex_Zeros
When finding the zeros of polynomials, at some point you’re faced with the problem x²=−1 . While there are clearly no real numbers that are solutions to this equation, leaving things there has a cert...When finding the zeros of polynomials, at some point you’re faced with the problem x²=−1 . While there are clearly no real numbers that are solutions to this equation, leaving things there has a certain feel of incompleteness. To address that, we will need utilize the imaginary unit, i .
4.E: Polynomial and Rational Functions (Exercises)
https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/04%3A_Polynomial_and_Rational_Functions/4.E%3A_Polynomial_and_Rational_Functions_(Exercises)
For the exercises 75-78, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. The $x$ -intercept is where the graph of...For the exercises 75-78, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. The $x$ -intercept is where the graph of the function crosses the $x$ -axis, and the zero of the function is the input value for which $f(x)=0$ .
1.4.E: Polynomial and Rational Functions (Exercises)
https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.04%3A_Polynomial_and_Rational_Functions/1.4.E%3A_Polynomial_and_Rational_Functions_(Exercises)
For the exercises 75-78, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. The $x$ -intercept is where the graph of...For the exercises 75-78, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. The $x$ -intercept is where the graph of the function crosses the $x$ -axis, and the zero of the function is the input value for which $f(x)=0$ .
3.R: Polynomial and Rational Functions(Review)
https://math.libretexts.org/Courses/Hartnell_College/MATH_25%3A_PreCalculus_(Abramson_OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.R%3A_Polynomial_and_Rational_Functions(Review)
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this...After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
3.1: Complex Numbers
https://math.libretexts.org/Courses/Hartnell_College/MATH_25%3A_PreCalculus_(Abramson_OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.01%3A_Complex_Numbers
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this...After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
3.2: Complex Numbers
https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/03%3A_Polynomial_and_Rational_Functions/3.02%3A_Complex_Numbers
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this...After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

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