Answer to Question #150810 in Algebra for kaila

Question

A) Create an example of a math problem that says to, “Factor fully” and that includes
Common Factoring of any polynomial.

B) Explain/Teach in detail what you had to do to arrive at your solution. Show all steps! l

Answer 1

$\displaystyle \textsf{Factorize}\,\, x^3 - 6x^2 + 11x - 6 = 0\\ \textsf{Let}\,\, f(x) = x^3 - 6x^2 + 11x - 6 = 0\\ \textsf{Substitute}\,\, x = 0, \pm1, \pm2,\cdots\\ f(0) = -6,\,\, f(1) = 1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0\\ \textsf{Since}\,\, f(1) = 0,\\ \textsf{By Factor theorem}\,\, x - 1\,\, \textsf{is a factor of the polynomial}\\ \textsf{Divide the polynomial}\\ \textsf{by the factor}\\ x^2 - 5x + 6\\ x - 1|x^3 - 6x^2 + 11x - 6 = 0\\ -(x^3 - x^2)\\ = -5x^2 + 11x - 6\\ -(-5x^2 + 5x)\\ = 6x - 6\\ -(6x - 6)\\ = 0\\ \textsf{Factorize the quotient}\\ x^2 - 5x + 6 = (x - 3)(x - 2)\\ f(x) = \textsf{divisor}\times \textsf{quotient}\\ \textsf{Therefore,}\,\,x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)$