Question

please factories this question

x³+ 13x² + 31x - 45
​

Answer 1

Step-by-step explanation:

Given -

• p(x) = x³ + 13x² + 31x - 45

To Find -

• Zeroes of the polynomial

Now,

→ x³ + 13x² + 31x - 45

→ x³ - x² + 14x² - 14x + 45x - 45

→ x²(x - 1)+ 14x(x - 1)+ 45(x - 1)

→ (x - 1)(x² + 14x + 45)

→ (x - 1)(x² + 5x + 9x + 45)

→ (x - 1)[x(x + 5) + 9(x + 5)]

→ (x - 1)(x + 9)(x + 5)

Zeroes are -

→ x - 1 = 0, x + 9 = 0 and x + 5 = 0

→ x = 1, x = -9, x = -5

Verification -

• αβγ = -d/a

→ 1 × -9 × -5 = -(-45)/1

→ 45 = 45

LHS = RHS

And

• α + β + γ = -b/a

→ 1 + (-9) + (-5) = -13/1

→ 1 - 9 - 5 = -13

→ -13 = -13

LHS = RHS

And

• αβ + βγ + γα = c/a

→ 1×-9 + -9×-5 + -5×1 = 31/1

→ -9 + 45 - 5 = 31

→ 31 = 31

LHS = RHS

Hence,

Verified..

Answer 2

$\large{\boxed{\underline{\underline{\bf{\red{Answer:-}}}}}}$

$\large\underline\mathrm{The \: value \: of \: x \: is \: 1, \: -9, \: -5.}$

$\large\underline\mathrm{Given:-}$

• p(x) = x³+ 13x² + 31x - 45

$\large\underline\mathrm{To \: find}$

• zeroes of the polynomial.

$\large\underline\mathrm{Solution}$

$\implies$ x³+ 13x² + 31x - 45

$\implies$ x³ - x² - 14x² + 14x + 45x - 45

$\implies$ x²(x - 1) + 14x(x - 1) + 45(x - 1)

$\implies$ (x - 1)(x² + 14x + 45)

$\implies$ (x - 1)(x² + 5x + 9x + 45)

$\implies$ (x - 1)[x(x + 5) + 9(x + 5)

$\implies$ (x - 1)(x + 5)(x + 9)

$\implies$ x - 1 = 0

$\implies$ x = 1

$\implies$ x + 9 = 0

$\implies$ x = -9

$\implies$ x + 5 = 0

$\implies$ x = -5

$\large\underline\mathrm{hence,}$

$\large\underline\mathrm{The \: value \: of \: x \: is \: 1, \: -9, \: -5.}$

$\large\underline\mathrm{Hope \: it \: helps \: you \: plz \: mark \: me \: brainlist}$