Question

➷ If x³ + y³ = 8 and x² + y² = 4, then find the value of x + y



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Answer 1

$\large\underline{\text{Main idea}}$

Polynomial identities contain sums or products of variables. So, if we substitute them with another, we can solve for their values.

$\large\underline{\text{Explanation}}$

$\large\boxed{\text{Part A: Products and sums}}$

Let us consider,

$\cdots \longrightarrow \begin{cases} & x+y=u \\ & xy=v \end{cases}.$

By polynomial identity,

$\cdots \longrightarrow x^3+y^3=(x+y)^3-3xy(x+y)$

$\cdots \longrightarrow \boxed{u^3-3uv=8.}$

By polynomial identity,

$\cdots \longrightarrow x^2+y^2=(x+y)^2-2xy$

$\cdots \longrightarrow \boxed{u^2-2v=4.}$

$\large\boxed{\text{Part B: Solving the equation}}$

The second equation for $v$ is,

$\cdots \longrightarrow 2v=u^2-4.$

By multiplying the first equation by 2,

$\cdots \longrightarrow 2u^3-6uv=16.$

Now let us substitute,

$\cdots \longrightarrow 2u^3-3u(u^2-4)=16$

$\cdots \longrightarrow -u^3+12u-16=0$

By substitution and synthetic division,

$\cdots\longrightarrow (u+4)^{2}(u-2)=0.$

Hence,

$\cdots \longrightarrow \text{ u=-4 (Double root) or u=2 .}$

And hence,

$\cdots \longrightarrow \text{ x+y=-4 or x+y=2 .}$

Answer 2

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