The polynomial px 3 + 4x 2 - 3x + q is completely divisible by x 2 - 1; find the values of p and q. Also for these values of p and q factorize the given polynomial completely.

x 2 - 1 is a factor of px 3 + 4x 2 - 3x + q. ∴ (x - 1) and (x + 1) are factors of px 3 + 4x 2 - 3x + q. Hence, substituting x = 1, -1 remainder = 0.. ⇒ p(1) 3 + 4(1) 2 - 3(1) + q = 0 ⇒ p + 4 - 3 + q = 0 ⇒ p + q = -1 ⇒ p = -1 - q .......(i) p(-1) 3 + 4(-1) 2 - 3(-1) + q = 0 ⇒ -p + 4 + 3 + q = 0 ⇒ p = 7 + q .......(ii) From (i) and (ii) we get, ⇒ -1 - q = 7 + q ⇒ 2q = -1 - 7 ⇒ 2q = -8 ⇒ q = -4. Substituting q = -4 in (i) we get, ⇒ p = -1 - (-4) = -1 + 4 = 3. Substituting p = 3 and q = -4 in px 3 + 4x 2 - 3x + q, = 3x 3 + 4x 2 - 3x - 4. On dividing, 3x 3 + 4x 2 - 3x - 4 by x - 1, we get, quotient = 3x 2 + 7x + 4 Factorising 3x 2 + 7x + 4, = 3x 2 + 3x + 4x + 4 = 3x(x + 1) + 4(x + 1) = (3x + 4)(x + 1). Hence, p = 3, q = -4 and 3x 3 + 4x 2 - 3x - 4 = (x - 1)(x + 1)(3x + 4).

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Mathematics

The polynomial px³ + 4x² - 3x + q is completely divisible by x² - 1; find the values of p and q. Also for these values of p and q factorize the given polynomial completely.

Factorisation

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Mathematics

The polynomial px³ + 4x² - 3x + q is completely divisible by x² - 1; find the values of p and q. Also for these values of p and q factorize the given polynomial completely.

Factorisation

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The polynomial px3 + 4x2 - 3x + q is completely divisible by x2 - 1; find the values of p and q. Also for these values of p and q factorize the given polynomial completely.

Factorisation

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