Solve (x+1)(x-1)(x+1) = x²+x³: Comparing Factored and Expanded Forms

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
00:03 We will use shortened multiplication formulas to open parentheses
00:13 1 squared is always equal to 1
00:23 Open parentheses properly
00:27 Each term multiplies each term
00:42 Collect terms and reduce what's possible
00:52 Isolate X
00:55 And this is the solution to the question

Step-by-Step Solution

Solve the equation by simplifying the expression on the left side in two stages. First, we'll multiply the expressions within the two leftmost pairs of parentheses:

Apply the shortened multiplication formula for squaring a binomial:

(a±b)2=a2±2ab+b2 (a\pm b)^2=a^2\pm2ab+b^2

Given that these two pairs of parentheses are being multiplied by another expression (which is also in parentheses), we'll place the result inside of parentheses (marked with an underline):

(x1)(x+1)(x+1)=x2+x3(x212)(x+1)=x2+x3(x21)(x+1)=x2+x3 \underline{ (x-1)(x+1)}(x+1)=x^2+x^3 \\ \underline{ (x^2-1^2)}(x+1)=x^2+x^3 \\ (x^2-1)(x+1)=x^2+x^3

Continue to simplify the expression on the left side by using the expanded distribution law:

(a+b)(c+d)=ac+ad+bc+bd (a+b)(c+d)=ac+ad+bc+bd

Additionally, we'll apply the law of exponents for multiplying terms with equal bases:

aman=am+n a^ma^n=a^{m+n}

Apply these laws in order to expand the parentheses in the expression in the equation:

(x21)(x+1)=x2+x3x3+x2x1=x2+x3 (x^2-1)(x+1)=x^2+x^3 \\ x^3+x^2-x-1=x^2+x^3 \\ Continue to combine like terms, while moving terms between sides. Later - we observe that the terms with squared and cubed powers cancel out, therefore it's a first-degree equation, which we'll solve by isolating the variable term and dividing both sides of the equation by its coefficient:

x3+x2x1=x2+x3x=1/:(1)x=1 x^3+x^2-x-1=x^2+x^3 \\ -x=1\hspace{8pt}\text{/}:(-1)\\ \boxed{x=-1}

Therefore, the correct answer is answer A.

Answer

x=1 x=-1