Question

Simplify (x2 + x - 1) (x²+ x +1 )​

Answer 1

Given: ($x^{2} +x-1$) ($x^{2} +x+1$)

To Find: Simplify the equation.

explanation:

we have,

                 $(x^{2} +x-1) (x^{2} +x+1)$

we can rearrange it into this,

                  $(x^{2} +x+1)(x^{2} +x-1)$

By applying the formula,

                   $(a+b) (a-b) = a^{2}-b^{2}$

Here, we will compare the given equation with the formula and by comparison we will have the values of a and b that will help to simplify the equation.

now,

                    $a = x^{2} +x$

                    $b = 1$

putting values of a and b into the formula,

                  $(x^{2}+x) ^{2} - (1)^{2}$

                  $x^{4} +x^{3} +x^{3} +x^{2} -1$

                   $x^{2} +2x^{3} +x^{2} -1$

hence our final product is $x^{2} +2x^{3} +x^{2} -1$

Answer 2

(x² + x - 1) (x²+ x +1 )​ = x⁴ + 2x³ + x²  - 1

Given:

• (x² + x - 1) (x²+ x +1 )​

To Find:

• Simplify the expression:

Solution:

• (a + b)² = a² + 2ab + b²
• (a + b)(a - b) = a² - b²
• (xᵃ)ᵇ = xᵃᵇ
• xᵃ.xᵇ=xᵃ⁺ᵇ

(x² + x - 1) (x²+ x +1 )​

Step 1:

Use (a - b)(a+ b) = a² - b² where a = x² + x  and b = 1

= (x² + x)² - 1²

Step 2:

Use (a + b)² = a² + 2ab + b²  where a = x²    and b = x  and 1² = 1

= (x²)² + 2(x²)(x) + x²  - 1

Step 3:

Use (x²)² = x⁴  and (x²)(x) = x³    
x⁴ + 2x³ + x²  - 1

Hence, (x² + x - 1) (x²+ x +1 )​ = x⁴ + 2x³ + x²  - 1