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Factorising quadratics when the coefficient of x squared ≠ 1 - Higher. Quadratic expressions can be written in the form \ (ax^2 + bx + c\), where \ (a\), \ (b\) and \ (c\) are numbers.
Every polynomial function with degree greater than 0 has at least one complex zero. Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\left(x-c\right)[/latex], where c is a complex number.
Nov 1, 2021 · When the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Synthetic division can be used to find the zeros of a polynomial function. According to the Fundamental Theorem, every polynomial function with degree greater than 0 has at least one complex zero.
Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
Use the Fundamental Theorem of Algebra to find complex zeros of a polynomial function. Use the Linear Factorization Theorem to find polynomials with given zeros. Use Descartes’ Rule of Signs to determine the maximum number of possible real zeros of a polynomial function. Solve real-world applications of polynomial equations.
Aug 15, 2024 · The Rational Zero Theorem states that, if the polynomial \(f(x)=a_nx^n+a_{n−1}x^{n−1}+...+a_1x+a_0\) has integer coefficients, then every rational zero of \(f(x)\) has the form \(\frac{p}{q}\) where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).
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The Fundamental Theorem of Algebra states that, if f (x) is a polynomial of degree n > 0, then f (x) has at least one complex zero. We can use this theorem to argue that, if f (x) is a polynomial of degree n> 0, and a is a non-zero real number, then f (x) has exactly n linear factors.
