If the zeroes of the polynomial x3 – 3x2 + x + 1 are (a – b), a an

1.	If the zeroes of the polynomial x3 – 3x2 + x + 1 are (a – b), a and (a + b), find the values of a and b.
Answer» The given polynomial = x³ – 3x² + x + 1 and its roots are (a – b), a and (a + b). Comparing the given polynomial with Ax³ + Bx² + Cx + D, we have: A = 1, B = -3, C = 1 and D = 1 Now, (a – b) + a + (a + b) = \(\frac{-B}A\) ⇒ 3 a = – \(\frac{-3}1\) ⇒ a = 1 Also, (a – b) × a × (a + b) = \(\frac{-D}A\) ⇒ a (a² – b²) = \(\frac{-1}1\) ⇒ 1 (1² – b² ) = -1 ⇒ 1– b² = -1 ⇒ b² = 2 ⇒ b = ±√2 ∴ a = 1 and b = ±√2

If the zeroes of the polynomial x3 – 3x2 + x + 1 are (a – b), a and (a + b), find the values of a and b.

Answer»

The given polynomial = x³ – 3x² + x + 1 and its roots are (a – b), a and (a + b).

Comparing the given polynomial with Ax³ + Bx² + Cx + D,

we have:

A = 1, B = -3, C = 1 and D = 1

Now, (a – b) + a + (a + b) = \(\frac{-B}A\)

⇒ 3 a = – \(\frac{-3}1\)

⇒ a = 1

Also, (a – b) × a × (a + b) = \(\frac{-D}A\)

⇒ a (a² – b²) = \(\frac{-1}1\)

⇒ 1 (1² – b² ) = -1

⇒ 1– b² = -1

⇒ b² = 2

⇒ b = ±√2

∴ a = 1 and b = ±√2