Question

the p(x) is a common multiple of degree 6 polynomial f(x) = x^3 -x^2-x-1 and g(x) = x^3-x^2+x-1​

Answer 1

Answer:

you have to do in the form of alpa and bith

-b/a

Answer 2

P(x) = x^6 - 2x^5 + x⁴ - 2x³ + x² - 1

P(x) is a common multiple of 6 degree polynomial , f(x) = x³ - x² - x - 1 and g(x) = x³ - x² + x - 1.

here we see f(x) and g(x) both are three degree polynomials if we multiply both the polynomials we will get 6 degree polynomial.

so, P(x) = f(x) × g(x)

= (x³ - x² - x - 1)(x³ - x² + x - 1)

= (x³ - x² - 1 - x)(x³ - x² - 1 + x)

from algebraic identities,

(a - b)(a + b) = a² - b²

so, P(x) = (x³ - x² - 1)² - x²

using formula, (a - b - c)² = a² + b² + c² - 2ab + 2bc - 2ca

= x^6 + x⁴ + 1 - 2x^5 + 2x² - 2x³ - x²

= x^6 - 2x^5 + x⁴ - 2x³ + x² - 1

hence, polynomial p(x) is x^6 - 2x^5 + x⁴ - 2x³ + x² - 1

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