https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-zeros/v/polynomial-zeros-grouping 2022-10-06T23:23:50.639907812Z Zeros of polynomials (with factoring): grouping When a polynomial is given in factored form, we can quickly find its zeros. When its given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor using the method of grouping. video https://cdn.kastatic.org/googleusercontent/iqDTC9bfViwTQSrnQICYjOCOzFgEG7UGBBczp2Aq4gwjrgXogT2eEOhUZSsPltn9nqdeBgSr-JG0EKJa7T62BN2l Zeros of polynomials (with factoring): grouping When a polynomial is given in factored form, we can quickly find its zeros. When its given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor using the method of grouping. https://cdn.kastatic.org/ka-youtube-converted/w0vTAHIS5B4.mp4/w0vTAHIS5B4.mp4 295 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-zeros/v/polynomial-zeros-common-factor 2022-10-06T23:23:50.639907812Z Zeros of polynomials (with factoring): common factor When a polynomial is given in factored form, we can quickly find its zeros. When it's given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor by first taking a common factor and then using the sum-product pattern. video https://cdn.kastatic.org/googleusercontent/0zIjZ0ASt2tv6tFK-GrZbcOHjU4J07rt0uQ8QU6LVoH3cPbE11TXWCsAwtxXRVdvnaVBfaBNtpZtx1cG4e35pOI Zeros of polynomials (with factoring): common factor When a polynomial is given in factored form, we can quickly find its zeros. When it's given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor by first taking a common factor and then using the sum-product pattern. https://cdn.kastatic.org/ka-youtube-converted/SONskIMkYdU.mp4/SONskIMkYdU.mp4 212 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-zeros/e/find-the-zeros-of-polynomials 2026-04-07T23:17:37.123726494Z Zeros of polynomials (with factoring) Use various methods in order to find all the zeros of polynomial expressions or functions. Tomer Gal exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-intervals/v/polynomial-multiplicity-examples 2022-10-06T23:23:50.639907812Z Zeros of polynomials (multiplicity) Given the graph of a polynomial and looking at its x-intercepts, we can determine the factors the polynomial must have. Additionally, we can determine whether those factors are raised to an odd power or to an even power (this is called the multiplicity of the factors). video https://cdn.kastatic.org/ka_thumbnails_cache/8128dbd2-af16-4742-a825-2f2c559e2496_1280_720_base.png Zeros of polynomials (multiplicity) Given the graph of a polynomial and looking at its x-intercepts, we can determine the factors the polynomial must have. Additionally, we can determine whether those factors are raised to an odd power or to an even power (this is called the multiplicity of the factors). https://cdn.kastatic.org/ka-youtube-converted/1ke_xceCevk.mp4/1ke_xceCevk.mp4 300 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-intervals/e/poly-zeros-mult 2026-04-30T21:00:44.23618726Z Zeros of polynomials (multiplicity) Relate polynomials to their graphs, where the polynomials have zeros with multiplicity greater than 1. Tomer Gal exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-eq/v/complex-roots-from-the-quadratic-formula 2023-08-25T17:33:50.463554978Z Solving quadratic equations: complex roots Solving quadratic equations can lead to complex solutions. Listen to Sal break down the process using the quadratic formula and standard form. Then, learn how to verify these complex solutions, ensuring you've got the right answer. It's all about turning tricky problems into solvable equations! Sal Khan video https://cdn.kastatic.org/googleusercontent/ohMf7xwqFsFM203jhcd1Hh8QJl-5jOzm67VmnhnuXowrNqSTNP-jRmDmOAck3wZbwXkeMWVavlj74TdJX5g74beJ Solving quadratic equations: complex roots Solving quadratic equations can lead to complex solutions. Listen to Sal break down the process using the quadratic formula and standard form. Then, learn how to verify these complex solutions, ensuring you've got the right answer. It's all about turning tricky problems into solvable equations! https://cdn.kastatic.org/ka-youtube-converted/dnjK4DPqh0k.mp4/dnjK4DPqh0k.mp4 615 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-eq/e/quadratic-formula-with-complex-solutions 2021-07-29T22:44:48.94944018Z Solve quadratic equations: complex solutions Solve quadratic equations using the quadratic formula. Some of the equations have real solutions while others have complex solutions. Tomer Gal exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:factor-high-deg/v/factor-high-deg-poly 2023-08-25T18:09:57.81238225Z Factoring higher degree polynomials Factoring higher degree polynomials involves breaking down complex expressions into simpler parts. This process includes identifying common factors, using the distributive property, and recognizing perfect squares. Sal demonstrates how to factor a partially factored polynomial and how to factor a third degree polynomial by grouping. video https://cdn.kastatic.org/googleusercontent/lsCu1uv79rcgXbhl7S5lWrDZ2s2bjG6KppkNWMvMfPdL3DLrj4x09bO5IQg-E9f1e-HcbUzgfK0VzdiMbyetCag6 Factoring higher degree polynomials Factoring higher degree polynomials involves breaking down complex expressions into simpler parts. This process includes identifying common factors, using the distributive property, and recognizing perfect squares. Sal demonstrates how to factor a partially factored polynomial and how to factor a third degree polynomial by grouping. https://cdn.kastatic.org/ka-youtube-converted/AUBX_aHStQY.mp4/AUBX_aHStQY.mp4 238 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:factor-high-deg/v/factoring-perfect-square-polynomial 2026-06-16T15:55:43.561813682Z Factoring higher degree polynomials: Common factor Discover the art of factoring higher degree polynomials. We start by pulling out common factors, then spot perfect squares. The key is seeing patterns and using them to simplify complex expressions. Sal demonstrates by factoring 16x^3+24x^2+9x as (x)(4x+3)^2. video https://cdn.kastatic.org/ka_thumbnails_cache/f901f348-de89-4c8f-87d9-c023e944a754_1280_720_base.png Factoring higher degree polynomials: Common factor Discover the art of factoring higher degree polynomials. We start by pulling out common factors, then spot perfect squares. The key is seeing patterns and using them to simplify complex expressions. Sal demonstrates by factoring 16x^3+24x^2+9x as (x)(4x+3)^2. https://cdn.kastatic.org/ka-youtube-converted/BFW2lHobO4E.mp4/BFW2lHobO4E.mp4 219 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:factor-high-deg/e/factor-higher-degree-polynomials 2026-06-16T16:34:33.656713573Z Factor higher degree polynomials Factor polynomials by taking common factors, grouping, and using known quadratic methods. Tomer Gal exercise https://www.khanacademy.org/math/ap-precalculus/xe322600f784e4778:polynomial-and-rational-functions/xe322600f784e4778:polynomial-functions-and-complex-zeros/v/applying-the-rational-root-theorem 2025-08-28T00:00:48.862639845Z Applying the rational root theorem Learn how to apply the rational root theorem to polynomial functions in order to solve for their zeros. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/7b4a3b53-b50f-4c29-97bf-3eb377f2c2e7_1280_720_base.png Applying the rational root theorem Learn how to apply the rational root theorem to polynomial functions in order to solve for their zeros. https://cdn.kastatic.org/ka-youtube-converted/Y9fwl_qocJc.mp4/Y9fwl_qocJc.mp4 455 Polynomial functions and complex zeros https://www.khanacademy.org/math/ap-precalculus/xe322600f784e4778:polynomial-and-rational-functions/xe322600f784e4778:polynomial-functions-and-complex-zeros/e/rational-root-theorem 2026-06-09T17:40:46.212160686Z Rational root theorem Identify possible zeros, confirm the zeros and find the remaining zeros of polynomials using the rational root theorem. Courtney Chernin-Hitchings exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:symmetry/v/even-and-odd-functions-equations 2022-10-06T23:23:50.639907812Z Even and odd functions: Equations When we are given the equation of a function f(x), we can check whether the function is even, odd, or neither by evaluating f(-x). If we get an expression that is equivalent to f(x), we have an even function; if we get an expression that is equivalent to -f(x), we have an odd function; and if neither happens, it is neither! video https://cdn.kastatic.org/ka_thumbnails_cache/2b0f5900-7483-4241-9383-4e8e99cd1f95_1280_720_base.png Even and odd functions: Equations When we are given the equation of a function f(x), we can check whether the function is even, odd, or neither by evaluating f(-x). If we get an expression that is equivalent to f(x), we have an even function; if we get an expression that is equivalent to -f(x), we have an odd function; and if neither happens, it is neither! https://cdn.kastatic.org/ka-youtube-converted/fTeO-V_VgEc.mp4/fTeO-V_VgEc.mp4 236 Polynomial functions and complex zeros https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:symmetry/e/determine-if-a-polynomial-is-even-or-odd 2025-06-25T21:21:38.768284711Z Even & odd functions: Equations Given the formula of a polynomial function, determine whether that function is even, odd, or neither. Tomer Gal exercise