Solve for x (1+x)(1-x+x^2-x^3+x^4)=1+a^5

$(1 + x) (1 - x + x^{2} - x^{3} + x^{4}) = 1 + a^{5}$

Step 1

Simplify $(1 + x) (1 - x + x^{2} - x^{3} + x^{4})$ .

Expand $(1 + x) (1 - x + x^{2} - x^{3} + x^{4})$ by multiplying each term in the first expression by each term in the second expression.

$1 \cdot 1 + 1 (- x) + 1 x^{2} + 1 (- x^{3}) + 1 x^{4} + x \cdot 1 + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Simplify terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 (- x) + 1 x^{2} + 1 (- x^{3}) + 1 x^{4} + x \cdot 1 + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $- x$ by $1$ .

$1 - x + 1 x^{2} + 1 (- x^{3}) + 1 x^{4} + x \cdot 1 + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x^{2}$ by $1$ .

$1 - x + x^{2} + 1 (- x^{3}) + 1 x^{4} + x \cdot 1 + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $- x^{3}$ by $1$ .

$1 - x + x^{2} - x^{3} + 1 x^{4} + x \cdot 1 + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x^{4}$ by $1$ .

$1 - x + x^{2} - x^{3} + x^{4} + x \cdot 1 + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x$ by $1$ .

$1 - x + x^{2} - x^{3} + x^{4} + x + x (- x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Rewrite using the commutative property of multiplication.

$1 - x + x^{2} - x^{3} + x^{4} + x - x \cdot x + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - (x \cdot x) + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x$ by $x$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x \cdot x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{1} x^{2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{1 + 2} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Add $1$ and $2$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} + x (- x^{3}) + x \cdot x^{4} = 1 + a^{5}$

Rewrite using the commutative property of multiplication.

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x \cdot x^{3} + x \cdot x^{4} = 1 + a^{5}$

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - (x^{3} x) + x \cdot x^{4} = 1 + a^{5}$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - (x^{3} x^{1}) + x \cdot x^{4} = 1 + a^{5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x^{3 + 1} + x \cdot x^{4} = 1 + a^{5}$

Add $3$ and $1$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x^{4} + x \cdot x^{4} = 1 + a^{5}$

Multiply $x$ by $x^{4}$ by adding the exponents.

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x^{4} + x^{1} x^{4} = 1 + a^{5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x^{4} + x^{1 + 4} = 1 + a^{5}$

Add $1$ and $4$ .

$1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x^{4} + x^{5} = 1 + a^{5}$

Combine the opposite terms in $1 - x + x^{2} - x^{3} + x^{4} + x - x^{2} + x^{3} - x^{4} + x^{5}$ .

Add $- x$ and $x$ .

$1 + x^{2} - x^{3} + x^{4} + 0 - x^{2} + x^{3} - x^{4} + x^{5} = 1 + a^{5}$

Add $1 + x^{2} - x^{3} + x^{4}$ and $0$ .

$1 + x^{2} - x^{3} + x^{4} - x^{2} + x^{3} - x^{4} + x^{5} = 1 + a^{5}$

Subtract $x^{2}$ from $x^{2}$ .

$1 - x^{3} + x^{4} + 0 + x^{3} - x^{4} + x^{5} = 1 + a^{5}$

Add $1 - x^{3} + x^{4}$ and $0$ .

$1 - x^{3} + x^{4} + x^{3} - x^{4} + x^{5} = 1 + a^{5}$

Add $- x^{3}$ and $x^{3}$ .

$1 + x^{4} + 0 - x^{4} + x^{5} = 1 + a^{5}$

Add $1 + x^{4}$ and $0$ .

$1 + x^{4} - x^{4} + x^{5} = 1 + a^{5}$

Subtract $x^{4}$ from $x^{4}$ .

$1 + 0 + x^{5} = 1 + a^{5}$

Add $1$ and $0$ .

$1 + x^{5} = 1 + a^{5}$

Step 2

Move all terms not containing $x$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$x^{5} = 1 + a^{5} - 1$

Combine the opposite terms in $1 + a^{5} - 1$ .

Subtract $1$ from $1$ .

$x^{5} = a^{5} + 0$

Add $a^{5}$ and $0$ .

$x^{5} = a^{5}$

Step 3

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$x = a$

Solve for x (1+x)(1-x+x^2-x^3+x^4)=1+a^5

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