Answer to Question #150811 in Algebra for kaila

Question #150811
A) Create an example of a math problem that says to, “Factor fully” and that includes
1
Expert's answer
2020-12-15T03:35:18-0500

Factorize  x36x2+11x6=0Let  f(x)=x36x2+11x6=0Substitute  x=0,±1,±2,f(0)=6,  f(1)=136(12)+11(1)6=16+116=0Since  f(1)=0,By Factor theorem  x1  is a factor of the polynomialDivide the polynomialby the factorx25x+6x1x36x2+11x6=0(x3x2)=5x2+11x6(5x2+5x)=6x6(6x6)=0Factorize the quotientx25x+6=(x3)(x2)f(x)=divisor×quotientTherefore,  x36x2+11x6=(x1)(x2)(x3)\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)


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