11th Grade
What Is the Fundamental Theorem of Algebra?
{
"voice_prompt": "",
"manuscript": {
"title": {
"text": "What Is the Fundamental Theorem of Algebra?",
"audio": "What is the Fundamental Theorem of Algebra?"
},
"description": {
"text": "The Fundamental Theorem of Algebra says that every polynomial equation of degree n has exactly n complex solutions, if we count repeated and imaginary ones too.",
"audio": "The Fundamental Theorem of Algebra says that every polynomial equation of degree n has exactly n complex solutions, if we count repeated and imaginary ones too."
},
"scenes": [
{
"text": "Let\u2019s say you have a polynomial P of x, defined as a sub n times x to the power of n, plus a, n minus 1 times x to the power of n minus 1, and so on, down to a, 1 times x plus a, 0.",
"latex": "\\( P(x) = a_nx^n + a_{n-1}x^{n-1} + \\dots + a_1x + a_0 \\)"
},
{
"text": "Then, according to the Fundamental Theorem of Algebra, the equation a, n times x to the power of n, plus a, n minus 1, times x to the power of n minus 1, and so on, equals zero \u2014 has exactly n complex solutions. That means it can be factored into a, n times x minus r, 1, times x minus r, 2, and so on, up to x minus r, n.",
"latex": "\\( a_nx^n + a_{n-1}x^{n-1} + \\dots + a_1x + a_0 = a_n(x - r_1)(x - r_2)\\dots(x - r_n) \\)"
},
{
"text": "Some of these solutions might be imaginary. For example, let's take a look at the equation x squared plus one equals zero. The polynomial has degree 2, and the equations has two solutions. You solve the equation x squared plus one equals zero, and get x equals i and x equals negative i. You have two complex solutions.",
"latex": "\\( x^2 + 1 = 0 \\implies x = \\pm i \\)"
},
{
"text": "Let's take a look at an example where the solutions are real. For example, if you solve x squared minus four equals zero, you get x equals two and x equals negative two. These are both real numbers.",
"latex": "\\( x^2 - 4 = 0 \\implies x = \\pm 2 \\)"
},
{
"text": "Sometimes, the same solution shows up more than once. For example, if you solve x minus two cubed equals zero, you get x equals two, repeated three times. You say that the solution has multiplicity of three.",
"latex": "\\( (x - 2)^3 = 0 \\implies x = 2 \\text{ with multiplicity } 3 \\)"
},
{
"text": "So no matter what kind of polynomial you have, the number of complex solutions always matches the degree of the polynomial. That\u2019s the power of the Fundamental Theorem of Algebra. It’s especially useful when you want to check whether you’ve found all the solutions to a polynomial equation. ",
"latex": "\\( \\text{Degree of the polynomial} = n \\implies \\text{Number of complex solutions} = n \\)"
}
],
"outro": {
"text": "So, what is the Fundamental Theorem of Algebra? It's the guarantee that every polynomial equation has exactly as many complex solutions as its degree, even if some of them are imaginary or repeated.",
"audio": "So, what is the Fundamental Theorem of Algebra? It's the guarantee that every polynomial equation has exactly as many complex solutions as its degree, even if some of them are imaginary or repeated."
}
}
}
en_11_alg_fund_theorem.json
Displaying en_11_alg_fund_theorem.json.
