Algebra

Advanced Factorization

Factorization of Polynomials

         

The polynomial 3x5+13x4+3x3+3x2+13x+33x^5+13x^4+3x^3+3x^2+13x+3 can be factorized as the product of two quadratics and one linear polynomial in xx, all with positive integer leading coefficients. What is the sum of all of the coefficients and constant terms in the three factors?

Given that f(x)f(x) and g(x)g(x) are two non-constant polynomials with integer coefficients such that

f(x)×g(x)=x66x5+4x412x38, f(x) \times g(x) = x^6-6x^5+4x^4-12x^3-8,

evaluate f(2)+g(2).\lvert f(2)+g(2) \rvert.

Details and assumptions

You may use the fact that f(2)×g(2)=168 f(2) \times g(2) = -168 .

For what constant kk can the polynomial (x+1)(x+5)(x+9)(x+13)+k(x+1)(x+5)(x+9)(x+13)+k be factorized into a perfect square of a quadratic in x?x?

If the polynomial x49x2+20x^4-9x^2+20 is factorized as (x+a)(x+b)(x+c)(x+d),(x+a)(x+b)(x+c)(x+d), where a,b,ca, b, c and dd are real numbers, what is the value of a2+b2+c2+d2?a^2+b^2+c^2+d^2?

Let f(x)=x21 f(x) = x^2 - 1 . How many distinct real roots are there to f(f(f(x)))=0 f ( f( f(x))) = 0 ?

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