Find the values of a and b, if (x - 1) and (x + 2) are both factors of (x³ + ax² + bx - 6).
Factorisation
6 Likes

Question

Find the values of a and b, if (x - 1) and (x + 2) are both factors of (x3 + ax2 + bx - 6).

KnowledgeBoat · Accepted Answer

Let f(x) = x3 + ax2 + bx - 6 Since (x − 1) and (x + 2) are factors, by the factor theorem, f(1) = 0 and f(−2) = 0. ⇒ f(1) = 0 ⇒ (1)3 + a(1)2 + b(1) - 6 = 0 ⇒ 1 + a + b - 6 = 0 ⇒ a + b - 5 = 0 ⇒ a + b = 5 ....(1) ⇒ f(-2) = 0 ⇒ (-2)3 + a(-2)2 + b(-2) - 6 = 0 ⇒ -8 + 4a - 2b - 6 = 0 ⇒ 4a - 2b = 14 ⇒ 2(2a - b) = 14 ⇒ 2a - b = ⇒ 2a - b = 7 ....(2) Adding equations (1) and (2), we get: ⇒ a + b + 2a - b = 5 + 7 ⇒ 3a = 12 ⇒ a = ⇒ a = 4. Substituting value of a in equation (1), we get : ⇒ 4 + b = 5 ⇒ b = 5 - 4 ⇒ b = 1. Hence, the value of a = 4 and b = 1.

Mathematics

Find the values of a and b, if (x - 1) and (x + 2) are both factors of (x³ + ax² + bx - 6).

Factorisation

Related Questions

If (x - 2) is a factor of 2x³ - x² - px - 2,
(i) find the value of p
(ii) with the value of p, factorize the above expression completely.

Find the value of a, if (x - a) is a factor of the polynomial 3x³ + x² - ax - 81.

If (x + 2) and (x + 3) are factors of x³ + ax + b, find the values of a and b.

If (x³ + ax² + bx + 6) has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find the values of a and b.

Mathematics

Find the values of a and b, if (x - 1) and (x + 2) are both factors of (x3 + ax2 + bx - 6).

Factorisation

Related Questions

If (x - 2) is a factor of 2x3 - x2 - px - 2,(i) find the value of p(ii) with the value of p, factorize the above expression completely.

Find the value of a, if (x - a) is a factor of the polynomial 3x3 + x2 - ax - 81.

If (x + 2) and (x + 3) are factors of x3 + ax + b, find the values of a and b.

If (x3 + ax2 + bx + 6) has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find the values of a and b.

Find the values of a and b, if (x - 1) and (x + 2) are both factors of (x³ + ax² + bx - 6).

If (x - 2) is a factor of 2x³ - x² - px - 2,
(i) find the value of p
(ii) with the value of p, factorize the above expression completely.

Find the value of a, if (x - a) is a factor of the polynomial 3x³ + x² - ax - 81.

If (x + 2) and (x + 3) are factors of x³ + ax + b, find the values of a and b.

If (x³ + ax² + bx + 6) has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find the values of a and b.