Question

If $x = 1 + \sqrt 2 $ then the value of expression ${x^4} - {x^3} - 2{x^2} - 3x + 1$ ?

Answer 1

Hint: A biquadratic equation is a 4-degree equation without the terms of degree 1 and 3. As the given value of x in the question, we will convert the given equation ${x^4} - {x^3} - 2{x^2} - 3x + 1$ in the form of a quadratic equation to get the value of the expression . then we will substitute the value of x

Complete answer:The value of $x = 1 + \sqrt 2 $
Breaking the equation in the form of quadratic equation
${x^4} - {x^3} - 2{x^2} - 3x + 1$
Taking ${x^2}$ common
$ \Rightarrow {x^2}({x^2} - x - 2) - 3x + 1$
Simplifying the equation ${x^2} - x - 2$
We gets $(x - 2)(x + 1)$
$ \Rightarrow {x^2}(x - 2)(x + 1) - 3x + 1$
Putting the value of x
$ \Rightarrow {(\sqrt 2 + 1)^2}(\sqrt 2 + 1 - 2)(\sqrt 2 + 1 + 1) - 3(\sqrt {2 + 1)} + 1$
$ \Rightarrow (\sqrt 2 + 1)(\sqrt 2 + 1)(\sqrt 2 - 1)(\sqrt 2 + 2) - 3(\sqrt 2 + 1) + 1$
Using $(a - b)(a + b)$ identity for $(\sqrt 2 + 1)(\sqrt 2 - 1)$
It will become ${(\sqrt 2 )^2} - {(1)^2}$ which will be $(2 - 1)$
$ \Rightarrow (\sqrt 2 + 1)(2 - 1)(\sqrt 2 + 2) - 3(\sqrt 2 + 1) + 1$
$ \Rightarrow (\sqrt 2 + 1)(\sqrt 2 + 2) - 3(\sqrt 2 + 1) + 1$
On further simplification the above expression
 ( $ + 3\sqrt 2 - 3\sqrt 2 $ ) will be canceled out
$ \Rightarrow 2 + 3\sqrt 2 + 2 - 3\sqrt 2 - 3 + 1$
 $ \Rightarrow 2$
The value of the expression is 2 .

Note:
Quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. The standard form of a quadratic equation is $a{x^2} + bx + c$ An interesting thing about quadratic equations is that they can have up to two real solutions. Solutions are where the quadratic equals 0. Real solutions mean that these solutions are not imaginary and are real numbers.
cubic equations are equations with a degree of 3. This means that the highest exponent is always 3 . The standard form of a cubic equation is $a{x^3} + b{x^2} + cx + d$ where a, b and c are the coefficients and d is the constant. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If the coefficients of a cubic equation are rational numbers, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple of their denominators.
Biquadratic equation is a 4-degree equation without the terms of degree 1 and 3. The standard form of biquadratic equation is $a{x^4} + b{x^3} + c{x^2} + dx + e$