Question

solve it plzzzzzzz ?????​

Answer 1

Answer:-

$\red{\bigstar}$$\large\leadsto\boxed{\bf\purple{a = 8}}$

$\red{\bigstar}$$\large\leadsto\boxed{\bf\purple{b = -8}}$

• Given:-

$\sf{p(x) = x^4 - 5x^3 + 4x^2 + ax + b}$

• To Find:-

Value of a and b

• Solution:-

Given that, x-1 and x-2 are the factors or roots of the given polynomial.

Hence,

→ $\sf{x - 1}$

$\implies\bf\red{x = 1}$

also,

→ $\sf{x - 2}$

$\implies\bf\red{x = 2}$

As these are the roots of the polynomial therefore they should satisfy the conditions as that of x.

$\pink{\bigstar}$ Taking $\bf\red{x = 1}$

Substituting the value in the polynomial:-

→ $\sf{(1)^4 - 5(1)^{3} + 4 (1)^{2} + a(1) + b = 0}$

→ $\sf{1 - 5 + 4 + a + b = 0}$

→ $\sf{5 -5 + a + b = 0}$

→ $\sf{a + b = 0}$

→ $\bf\green{a = - b}$

$\pink{\bigstar}$ Taking $\bf\red{x = 2}$

Substituting the value in the polynomial:-

→ $\sf{(2)^4 - 5(2)^{3} + 4 (2)^{2} + a(2) + b = 0}$

→ $\sf{16 - 5 \times 8 + 4 \times 4 + 2a + b = 0}$

→ $\sf{16 - 40 + 16 + 2a + b = 0}$

→ $\sf{32 - 40 + 2a + b = 0}$

→ $\sf{-8 + 2a + b = 0}$

→ $\sf{2a + b = 8}$

→ $\sf{2(-b) + b = 8}$$\: \: \: \: \:\longrightarrow\bf\red{[a = -b]}$

→ $\sf{-2b + b = 8}$

→ $\sf{-b = 8}$

→ $\bf\green{b = -8}$

Therefore,

$\large\boxed{\bf\pink{b = -8}}$

And

As, a = -b

Hence,

→ a = -(-8)

$\large\boxed{\bf\pink{a = 8}}$

Answer 2

$\huge{\underline{\underline{\boxed{\sf{\purple{Answer ࿐}}}}}}$

Since (1 – √5) is a root of the polynomial P(x) = 0 (1 + √5) is also a root of P (x) = 0

⇒ x2 – [(1 + √5) + (1 – √5)]x + (1 + √5) (1 – √5) = 0 is a factor of P(x) = 0

⇒ x2 – 2x – 4 = 0 is a factor of P(x) = 0.

Dividing the polynomial by x2 – 2x – 4 = 0 We get the other factor x2 – 3x + 2 = 0

The roots of x2 – 3x + 2 = 0

(x – 2) (x – 1) = 0

x = 1, 2

The roots are 1, 2, 1 ± √5