Given that x - 2 and x + 1 are factors of f(x) = x 3 + 3x 2 + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

x - 2 = 0 ⇒ x = 2. Since, x - 2 is a factor of x 3 + 3x 2 + ax + b. Hence, on substituting x = 2 in above expression, remainder = 0. ⇒ (2) 3 + 3(2) 2 + a(2) + b = 0 ⇒ 8 + 12 + 2a + b = 0 ⇒ 2a + b = -20 ⇒ b = -20 - 2a ........(i) x + 1 = 0 ⇒ x = -1. Since, x + 1 is a factor of x 3 + 3x 2 + ax + b. Hence, on substituting x = -1 in above expression, remainder = 0. ⇒ (-1) 3 + 3(-1) 2 + a(-1) + b = 0 ⇒ -1 + 3 - a + b = 0 ⇒ 2 - a + b = 0 ⇒ b = a - 2 .......(ii) From (i) and (ii) we get, ⇒ -20 - 2a = a - 2 ⇒ a + 2a = -20 + 2 ⇒ 3a = -18 ⇒ a = -6. Substituting value of a in (ii) we get, ⇒ b = a - 2 = -6 - 2 = -8. ∴ a = -6 and b = -8. On dividing, x 3 + 3x 2 - 6x - 8 by (x - 2), we get, quotient = x 2 + 5x + 4. Factorising x 2 + 5x + 4, = x 2 + 4x + x + 4 = x(x + 4) + 1(x + 4) = (x + 1)(x + 4). Hence, a = -6, b = -8 and f(x) = (x - 2)(x + 1)(x + 4).

Using the Remainder Theorem, factorise the expression 3x³ + 10x² + x - 6. Hence, solve the equation 3x³ + 10x² + x - 6 = 0.

View Answer Bookmark Now

Factorise the expression
f(x) = 2x³ - 7x² - 3x + 18.
Hence, find all possible values of x for which f(x) = 0.

View Answer Bookmark Now

The expression 4x³ - bx² + x - c leaves remainders 0 and 30 when divided by x + 1 and 2x - 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.

View Answer Bookmark Now

Mathematics

Given that x - 2 and x + 1 are factors of f(x) = x³ + 3x² + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

Factorisation

Related Questions

Using the Remainder Theorem, factorise the expression 3x³ + 10x² + x - 6. Hence, solve the equation 3x³ + 10x² + x - 6 = 0.

Factorise the expression
f(x) = 2x³ - 7x² - 3x + 18.
Hence, find all possible values of x for which f(x) = 0.

The expression 4x³ - bx² + x - c leaves remainders 0 and 30 when divided by x + 1 and 2x - 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.

Mathematics

Given that x - 2 and x + 1 are factors of f(x) = x3 + 3x2 + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

Factorisation

Related Questions

Using the Remainder Theorem, factorise the expression 3x3 + 10x2 + x - 6. Hence, solve the equation 3x3 + 10x2 + x - 6 = 0.

Factorise the expressionf(x) = 2x3 - 7x2 - 3x + 18.Hence, find all possible values of x for which f(x) = 0.

The expression 4x3 - bx2 + x - c leaves remainders 0 and 30 when divided by x + 1 and 2x - 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.

Given that x - 2 and x + 1 are factors of f(x) = x³ + 3x² + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

Using the Remainder Theorem, factorise the expression 3x³ + 10x² + x - 6. Hence, solve the equation 3x³ + 10x² + x - 6 = 0.

Factorise the expression
f(x) = 2x³ - 7x² - 3x + 18.
Hence, find all possible values of x for which f(x) = 0.

The expression 4x³ - bx² + x - c leaves remainders 0 and 30 when divided by x + 1 and 2x - 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.