Polynomial long division works similarly to long division with numbers. We divide the leading terms, multiply, subtract, and repeat until we reach the end of the dividend.<\/p>\r\n
The process:<\/p>\r\n\r\n
For polynomial dividend<\/strong> [latex]f(x)[\/latex] and non-zero divisor<\/strong> [latex]d(x)[\/latex]:<\/p>\r\n [latex]f(x) = d(x)q(x) + r(x)[\/latex]<\/p>\r\n where:<\/p>\r\n\r\n Synthetic division is a shortcut for dividing polynomials by binomials of the form [latex]x - k[\/latex].<\/p>\r\n Steps:<\/p>\r\n\r\n Imaginary Numbers<\/strong><\/p>\r\n The imaginary unit [latex]i[\/latex] is defined as [latex]i = \\sqrt{-1}[\/latex], which means [latex]i^2 = -1[\/latex].<\/p>\r\n Any square root of a negative number can be written as a multiple of [latex]i[\/latex]: [latex]\\sqrt{-25} = \\sqrt{25} \\cdot \\sqrt{-1} = 5i[\/latex]<\/p>\r\n Complex Numbers<\/strong><\/p>\r\n A complex number has the form [latex]a + bi[\/latex] where:<\/p>\r\n\r\n The Complex Plane<\/strong><\/p>\r\n Complex numbers are plotted on a coordinate system where:<\/p>\r\n\r\n Addition and Subtraction:<\/p>\r\n Combine real parts and imaginary parts separately<\/p>\r\n\r\n Multiplication:<\/p>\r\n Use the distributive property (FOIL), remembering that [latex]i^2 = -1[\/latex]<\/p>\r\n\r\n Multiplying by a real number:<\/p>\r\n Distribute to both parts<\/p>\r\n\r\n The complex conjugate of [latex]a + bi[\/latex] is [latex]a - bi[\/latex].<\/p>\r\n When you multiply complex conjugates, the result is always a real number:<\/p>\r\n\r\n The powers of [latex]i[\/latex] cycle in a pattern of 4:<\/p>\r\n\r\n Remainder Theorem<\/strong><\/p>\r\n If a polynomial [latex]f(x)[\/latex] is divided by [latex]x - k[\/latex], then the remainder is [latex]f(k)[\/latex].<\/p>\r\n Factor Theorem<\/strong><\/p>\r\n [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x - k)[\/latex] is a factor of [latex]f(x)[\/latex].<\/p>\r\n This means:<\/p>\r\n\r\n Rational Zero Theorem<\/strong><\/p>\r\n For polynomial [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_1x + a_0[\/latex] with integer coefficients:<\/p>\r\n Every possible rational zero has the form [latex]\\frac{p}{q}[\/latex] where:<\/p>\r\n\r\n Steps:<\/p>\r\n\r\n Fundamental Theorem of Algebra<\/strong><\/p>\r\n Every polynomial function of degree [latex]n > 0[\/latex] has at least one complex zero.<\/p>\r\n This means:<\/p>\r\n\r\n Linear Factorization Theorem<\/strong><\/p>\r\n A polynomial function of degree [latex]n[\/latex] can be written as a product of [latex]n[\/latex] linear factors:<\/p>\r\n [latex]f(x) = a(x - c_1)(x - c_2) \\cdots (x - c_n)[\/latex]<\/p>\r\n where [latex]c_1, c_2, \\ldots, c_n[\/latex] are complex numbers (possibly including real numbers).<\/p>\r\n Complex Conjugate Theorem<\/strong><\/p>\r\n If a polynomial function [latex]f[\/latex] has real coefficients and [latex]a + bi[\/latex] (where [latex]b \\ne 0[\/latex]) is a zero, then the complex conjugate [latex]a - bi[\/latex] must also be a zero.<\/p>\r\n This means complex zeros always come in conjugate pairs for polynomials with real coefficients.<\/em><\/p>\r\n\r\n Given zeros and a point on the graph:<\/p>\r\n\r\n To solve inequalities like [latex]f(x) > 0[\/latex] or [latex]f(x) < 0[\/latex]:<\/p>\r\n Method 1 (Test Values):<\/p>\r\n\r\n Method 2 (Graphing):<\/p>\r\n\r\n Not all polynomial functions have inverses that are functions. A polynomial must be one-to-one (pass the horizontal line test) to have an inverse function.<\/p>\r\n Notation: [latex]f^{-1}(x)[\/latex] denotes the inverse function, NOT [latex]\\frac{1}{f(x)}[\/latex]<\/em><\/p>\r\n Finding Inverses of One-to-One Polynomials<\/strong><\/p>\r\n Steps:<\/p>\r\n\r\n Key properties:<\/p>\r\n\r\n Restricting the Domain<\/strong><\/p>\r\n When a polynomial is not one-to-one (like quadratic functions), we can restrict its domain to make it one-to-one, then find the inverse on that restricted domain.<\/p>\r\n For [latex]f(x) = (x - h)^2 + k[\/latex]:<\/p>\r\n\r\n The outputs of the inverse must match the restricted domain of the original function.<\/p>\r\n\r\n complex conjugate<\/strong><\/p>\r\n the complex number in which the sign of the imaginary part is changed and the real part is left unchanged; when multiplied by or added to the original complex number, the result is a real number<\/p>\r\n complex number<\/strong><\/p>\r\n a number of the form [latex]a + bi[\/latex] where [latex]a[\/latex] is the real part and [latex]bi[\/latex] is the imaginary part<\/p>\r\n complex plane<\/strong><\/p>\r\n a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number<\/p>\r\n Descartes' Rule of Signs<\/strong><\/p>\r\n a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f(x)[\/latex] and [latex]f(-x)[\/latex]<\/p>\r\n Division Algorithm<\/strong><\/p>\r\n given polynomial dividend [latex]f(x)[\/latex] and non-zero polynomial divisor [latex]d(x)[\/latex] where the degree of [latex]d(x)[\/latex] is less than or equal to the degree of [latex]f(x)[\/latex], there exist unique polynomials [latex]q(x)[\/latex] and [latex]r(x)[\/latex] such that [latex]f(x) = d(x)q(x) + r(x)[\/latex]<\/p>\r\n Factor Theorem<\/strong><\/p>\r\n [latex]k[\/latex] is a zero of polynomial function [latex]f(x)[\/latex] if and only if [latex](x - k)[\/latex] is a factor of [latex]f(x)[\/latex]<\/p>\r\n Fundamental Theorem of Algebra<\/strong><\/p>\r\n a polynomial function with degree greater than 0 has at least one complex zero<\/p>\r\n imaginary number<\/strong><\/p>\r\n a number in the form [latex]bi[\/latex] where [latex]i = \\sqrt{-1}[\/latex]<\/p>\r\n invertible function<\/strong><\/p>\r\n any function that has an inverse function<\/p>\r\n Linear Factorization Theorem<\/strong><\/p>\r\n allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex](x - c)[\/latex] where [latex]c[\/latex] is a complex number<\/p>\r\n Rational Zero Theorem<\/strong><\/p>\r\n the possible rational zeros of a polynomial function have the form [latex]\\frac{p}{q}[\/latex] where [latex]p[\/latex] is a factor of the constant term and [latex]q[\/latex] is a factor of the leading coefficient<\/p>\r\n Remainder Theorem<\/strong><\/p>\r\n if a polynomial [latex]f(x)[\/latex] is divided by [latex]x - k[\/latex], then the remainder is equal to the value [latex]f(k)[\/latex]<\/p>\r\n synthetic division<\/strong><\/p>\r\n a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x - k[\/latex]<\/p>","rendered":" Polynomial long division works similarly to long division with numbers. We divide the leading terms, multiply, subtract, and repeat until we reach the end of the dividend.<\/p>\n The process:<\/p>\n For polynomial dividend<\/strong> [latex]f(x)[\/latex] and non-zero divisor<\/strong> [latex]d(x)[\/latex]:<\/p>\n [latex]f(x) = d(x)q(x) + r(x)[\/latex]<\/p>\n where:<\/p>\n Synthetic division is a shortcut for dividing polynomials by binomials of the form [latex]x - k[\/latex].<\/p>\n Steps:<\/p>\n Imaginary Numbers<\/strong><\/p>\n The imaginary unit [latex]i[\/latex] is defined as [latex]i = \\sqrt{-1}[\/latex], which means [latex]i^2 = -1[\/latex].<\/p>\n Any square root of a negative number can be written as a multiple of [latex]i[\/latex]: [latex]\\sqrt{-25} = \\sqrt{25} \\cdot \\sqrt{-1} = 5i[\/latex]<\/p>\n Complex Numbers<\/strong><\/p>\n A complex number has the form [latex]a + bi[\/latex] where:<\/p>\n The Complex Plane<\/strong><\/p>\n Complex numbers are plotted on a coordinate system where:<\/p>\n Addition and Subtraction:<\/p>\n Combine real parts and imaginary parts separately<\/p>\n Multiplication:<\/p>\n Use the distributive property (FOIL), remembering that [latex]i^2 = -1[\/latex]<\/p>\n Multiplying by a real number:<\/p>\n Distribute to both parts<\/p>\n The complex conjugate of [latex]a + bi[\/latex] is [latex]a - bi[\/latex].<\/p>\n When you multiply complex conjugates, the result is always a real number:<\/p>\n The powers of [latex]i[\/latex] cycle in a pattern of 4:<\/p>\n Remainder Theorem<\/strong><\/p>\n If a polynomial [latex]f(x)[\/latex] is divided by [latex]x - k[\/latex], then the remainder is [latex]f(k)[\/latex].<\/p>\n Factor Theorem<\/strong><\/p>\n [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x - k)[\/latex] is a factor of [latex]f(x)[\/latex].<\/p>\n This means:<\/p>\n Rational Zero Theorem<\/strong><\/p>\n For polynomial [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_1x + a_0[\/latex] with integer coefficients:<\/p>\n Every possible rational zero has the form [latex]\\frac{p}{q}[\/latex] where:<\/p>\n Steps:<\/p>\n Fundamental Theorem of Algebra<\/strong><\/p>\n Every polynomial function of degree [latex]n > 0[\/latex] has at least one complex zero.<\/p>\n This means:<\/p>\n Linear Factorization Theorem<\/strong><\/p>\n A polynomial function of degree [latex]n[\/latex] can be written as a product of [latex]n[\/latex] linear factors:<\/p>\n [latex]f(x) = a(x - c_1)(x - c_2) \\cdots (x - c_n)[\/latex]<\/p>\n where [latex]c_1, c_2, \\ldots, c_n[\/latex] are complex numbers (possibly including real numbers).<\/p>\n Complex Conjugate Theorem<\/strong><\/p>\n If a polynomial function [latex]f[\/latex] has real coefficients and [latex]a + bi[\/latex] (where [latex]b \\ne 0[\/latex]) is a zero, then the complex conjugate [latex]a - bi[\/latex] must also be a zero.<\/p>\n This means complex zeros always come in conjugate pairs for polynomials with real coefficients.<\/em><\/p>\n Given zeros and a point on the graph:<\/p>\n To solve inequalities like [latex]f(x) > 0[\/latex] or [latex]f(x) < 0[\/latex]:<\/p>\n Method 1 (Test Values):<\/p>\n Method 2 (Graphing):<\/p>\n Not all polynomial functions have inverses that are functions. A polynomial must be one-to-one (pass the horizontal line test) to have an inverse function.<\/p>\n Notation: [latex]f^{-1}(x)[\/latex] denotes the inverse function, NOT [latex]\\frac{1}{f(x)}[\/latex]<\/em><\/p>\n Finding Inverses of One-to-One Polynomials<\/strong><\/p>\n Steps:<\/p>\n Key properties:<\/p>\n Restricting the Domain<\/strong><\/p>\n When a polynomial is not one-to-one (like quadratic functions), we can restrict its domain to make it one-to-one, then find the inverse on that restricted domain.<\/p>\n For [latex]f(x) = (x - h)^2 + k[\/latex]:<\/p>\n The outputs of the inverse must match the restricted domain of the original function.<\/p>\n complex conjugate<\/strong><\/p>\n the complex number in which the sign of the imaginary part is changed and the real part is left unchanged; when multiplied by or added to the original complex number, the result is a real number<\/p>\n complex number<\/strong><\/p>\n a number of the form [latex]a + bi[\/latex] where [latex]a[\/latex] is the real part and [latex]bi[\/latex] is the imaginary part<\/p>\n complex plane<\/strong><\/p>\n a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number<\/p>\n Descartes’ Rule of Signs<\/strong><\/p>\n a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f(x)[\/latex] and [latex]f(-x)[\/latex]<\/p>\n Division Algorithm<\/strong><\/p>\n given polynomial dividend [latex]f(x)[\/latex] and non-zero polynomial divisor [latex]d(x)[\/latex] where the degree of [latex]d(x)[\/latex] is less than or equal to the degree of [latex]f(x)[\/latex], there exist unique polynomials [latex]q(x)[\/latex] and [latex]r(x)[\/latex] such that [latex]f(x) = d(x)q(x) + r(x)[\/latex]<\/p>\n Factor Theorem<\/strong><\/p>\n [latex]k[\/latex] is a zero of polynomial function [latex]f(x)[\/latex] if and only if [latex](x - k)[\/latex] is a factor of [latex]f(x)[\/latex]<\/p>\n Fundamental Theorem of Algebra<\/strong><\/p>\n a polynomial function with degree greater than 0 has at least one complex zero<\/p>\n imaginary number<\/strong><\/p>\n a number in the form [latex]bi[\/latex] where [latex]i = \\sqrt{-1}[\/latex]<\/p>\n invertible function<\/strong><\/p>\n any function that has an inverse function<\/p>\n Linear Factorization Theorem<\/strong><\/p>\n allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex](x - c)[\/latex] where [latex]c[\/latex] is a complex number<\/p>\n Rational Zero Theorem<\/strong><\/p>\n\r\n \t
Synthetic Division<\/strong><\/h4>\r\n
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Complex Numbers<\/h3>\r\n
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Arithmetic with Complex Numbers<\/strong><\/h4>\r\n
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Complex Conjugates<\/strong><\/h4>\r\n
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Powers of [latex]i[\/latex]<\/strong><\/h4>\r\n
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Zeros of Polynomial Functions<\/h3>\r\n
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Finding Zeros of Polynomial Functions<\/strong><\/h4>\r\n
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Writing Polynomials from Zeros<\/strong><\/h4>\r\n
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Polynomial Inequalities and Inverses<\/h3>\r\n
Solving Polynomial Inequalities<\/strong><\/h4>\r\n
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Inverse of Polynomial Functions<\/strong><\/h4>\r\n
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Key Equations<\/h2>\r\n
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\r\n Imaginary unit<\/strong><\/td>\r\n [latex]i = \\sqrt{-1}[\/latex], therefore [latex]i^2 = -1[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Complex number standard form<\/strong><\/td>\r\n [latex]a + bi[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n Glossary<\/h2>\r\n
Essential Concepts<\/h2>\n
Dividing Polynomials<\/h3>\n
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The Division Algorithm<\/strong><\/h4>\n
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Synthetic Division<\/strong><\/h4>\n
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Complex Numbers<\/h3>\n
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Arithmetic with Complex Numbers<\/strong><\/h4>\n
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Complex Conjugates<\/strong><\/h4>\n
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Powers of [latex]i[\/latex]<\/strong><\/h4>\n
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Zeros of Polynomial Functions<\/h3>\n
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Finding Zeros of Polynomial Functions<\/strong><\/h4>\n
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Writing Polynomials from Zeros<\/strong><\/h4>\n
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Polynomial Inequalities and Inverses<\/h3>\n
Solving Polynomial Inequalities<\/strong><\/h4>\n
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Inverse of Polynomial Functions<\/strong><\/h4>\n
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Key Equations<\/h2>\n
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\n Imaginary unit<\/strong><\/td>\n [latex]i = \\sqrt{-1}[\/latex], therefore [latex]i^2 = -1[\/latex]<\/td>\n<\/tr>\n \n Complex number standard form<\/strong><\/td>\n [latex]a + bi[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n Glossary<\/h2>\n