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The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms.
For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. [ 6 ] : 202–207 If one is given a quadratic equation in the form x 2 + bx + c = 0 , the sought factorization has the form ( x + q )( x + s ) , and one has to find two numbers q and s that add up to b and whose product is c ...
If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
A finite-dimensional vector space with a quadratic form is called a quadratic space. The map Q is a homogeneous function of degree 2, which means that it has the property that, for all a in K and v in V : Q ( a v ) = a 2 Q ( v ) . {\displaystyle Q(av)=a^{2}Q(v).}
A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
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