Solve (x+1)(x-1)(x+1) = x^3 + x^2: Cubic Equation with Repeated Factor

Question

Solve the following problem:

(x+1)(x1)(x+1)=x2+x3 (x+1)(x-1)(x+1)=x^2+x^3

Video Solution

Solution Steps

00:00 Solve
00:09 Extract a common factor from the parentheses
00:16 Reduce what's possible
00:32 Use the shortened multiplication formulas to open the parentheses
00:40 Reduce what's possible
00:44 We got an illogical expression
00:49 Let's check if we divided by 0, which is why we got an illogical expression
00:52 Let's check if X equals minus 1, meaning we divided by 0
00:57 Let's substitute -1 in the exercise and solve
01:14 Now we got a logical expression, so this is the solution for X
01:18 And this is the solution to the question

Step-by-Step Solution

Let's examine the given equation:

(x+1)(x1)(x+1)=x2+x3 (x+1)(x-1)(x+1)=x^2+x^3

Begin by opening the second and third pairs of parentheses from the left (marked with an underline below) On the left side we can apply the difference of squares formula:

(a+b)(ab)=a2b2 (a+b)(a-b)=a^2-b^2 ,

Place the result inside of new parentheses (since the resulting expression in its entirety is multiplied by an expression that is enclosed by these parentheses) then we'll proceed to simplify the expression in the resulting parentheses:

(x+1)(x1)(x+1)=x2+x3(x+1)(x212)=x2+x3(x+1)(x21)=x2+x3 (x+1)\underline{(x-1)(x+1)}=x^2+x^3 \\ \downarrow\\ (x+1)\underline{\textcolor{blue}{(}x^2-1^2\textcolor{blue}{)}}=x^2+x^3\\ (x+1)\underline{\textcolor{blue}{(}x^2-1\textcolor{blue}{)}}=x^2+x^3\\ Continue to use the expanded distributive law again and open the parentheses on the left side. Proceed to move and combine like terms:

(x+1)(x21)=x2+x3x3x+x21=x2+x3x=1/(1)x=1 (x+1)(x^2-1)=x^2+x^3\\ \downarrow\\ x^3-x+x^2-1=x^2+x^3\\ -x =1\hspace{6pt}\text{/}\cdot(-1)\\ \downarrow\\ \boxed{x=-1}

In the final step solve the first-degree equation that we obtained,

Therefore the correct answer is answer A.

Answer

x=1 x=-1