Given that $2+\sqrt5$ is a root of $x^3-2x^2-9x-2=0$, find all the roots.

Given that 2+√5 is a root of x³-2x²-9x-2=0, find all the roots.

Exercise 4(c) · Q3

2+\sqrt5

Telangana TsbieTextbookSubjectiveImportance★★★★★est

31% · 11/36 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Step 1. Since $f(x)=x^3-2x^2-9x-2$ has rational coefficients and $2+\sqrt5$ is a root, its conjugate $2-\sqrt5$ is also a root (Irrational Conjugate Roots theorem).

Step 2. These contribute the rational quadratic factor

\big(x-(2+\sqrt5)\big)\big(x-(2-\sqrt5)\big)=(x-2)^2-5=x^2-4x+4-5=x^2-4x-1.

Step 3. Divide $f(x)$ by $x^2-4x-1$ :

$x^3-2x^2-9x-2 = (x^2-4x-1)(x+2) + 0,$ …

Start your 14-day free trial to unlock the full solution →

←Previous question · 4.3.2Q2 — Given that 1+2i is a root of x⁴-4x³+6x²-4x-15=0, find all the roots.→Next question · 4.3.2Q4 — Form the monic cubic equation with rational coefficients having 3-√2 a…

Q.Given that $2+\sqrt5$ is a root of $x^3-2x^2-9x-2=0$ , find all the roots.

Concept understanding — Irrational Conjugate Roots

Q.Given that 2+52+\sqrt52+5​ is a root of x3−2x2−9x−2=0x^3-2x^2-9x-2=0x3−2x2−9x−2=0, find all the roots.

🎓Unlock everything free for 14 days

Q.Given that $2+\sqrt5$ is a root of $x^3-2x^2-9x-2=0$ , find all the roots.

Unlock everything free for 14 days