If (x + 3) and (x - 4) are factors of x 3 + ax 2 - bx + 24, find the values of a and b. With these values of a and b, factorise the given expression.

By factor theorem (x - b) is a factor of f(x), if f(b) = 0. f(x) = x 3 + ax 2 - bx + 24 Given, (x + 3) or (x - (-3) and (x - 4) are factors of f(x) ∴ f(-3) = 0 and f(4) = 0. For, f(-3) = 0 On dividing equation by 3, For f(4) = 0 On dividing equation by 4, Putting value of b = 1 - 3a from equation 1, Now putting a = -3 and b = 10 in f(x), f(x) = x 3 - 3x 2 - 10x + 24 Since, (x + 3) and (x - 4) is a factor of f(x), hence (x + 3)(x + 4) is also the factor On dividing, f(x) by x 2 - x - 12, we get (x - 2) as quotient and remainder = 0. Hence, value of a = -3 and b = 10; x 3 - 3x 2 - 10x + 24 = (x - 2)(x + 3)(x - 4).

Use factor theorem to factorise the following polynomials completely :
(i) 4x³ + 4x² - 9x - 9
(ii) x³ - 19x - 30
(iii) 2x³ - x² - 13x - 6

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If x³ - 2x² + px + q has a factor (x + 2) and leaves a remainder 9 when divided by (x + 1), find the values of p and q. With these values of p and q, factorise the given polynomial completely.

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If (2x + 1) is a factor of both the expressions 2x² - 5x + p and 2x² + 5x + q, find the values of p and q. Hence, find the other factors of both the polynomials.

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If a polynomial f(x) = x⁴ - 2x³ + 3x² - ax - b leaves remainders 5 and 19 when divided by (x - 1) and (x + 1) respectively, find the values of a and b. Hence, determine the remainder when f(x) is divided by (x - 2).

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Mathematics

If (x + 3) and (x - 4) are factors of x³ + ax² - bx + 24, find the values of a and b. With these values of a and b, factorise the given expression.

Factorisation

Related Questions

Use factor theorem to factorise the following polynomials completely :
(i) 4x³ + 4x² - 9x - 9
(ii) x³ - 19x - 30
(iii) 2x³ - x² - 13x - 6

If x³ - 2x² + px + q has a factor (x + 2) and leaves a remainder 9 when divided by (x + 1), find the values of p and q. With these values of p and q, factorise the given polynomial completely.

If (2x + 1) is a factor of both the expressions 2x² - 5x + p and 2x² + 5x + q, find the values of p and q. Hence, find the other factors of both the polynomials.

If a polynomial f(x) = x⁴ - 2x³ + 3x² - ax - b leaves remainders 5 and 19 when divided by (x - 1) and (x + 1) respectively, find the values of a and b. Hence, determine the remainder when f(x) is divided by (x - 2).

Mathematics

If (x + 3) and (x - 4) are factors of x3 + ax2 - bx + 24, find the values of a and b. With these values of a and b, factorise the given expression.

Factorisation

Related Questions

Use factor theorem to factorise the following polynomials completely :(i) 4x3 + 4x2 - 9x - 9(ii) x3 - 19x - 30(iii) 2x3 - x2 - 13x - 6

If x3 - 2x2 + px + q has a factor (x + 2) and leaves a remainder 9 when divided by (x + 1), find the values of p and q. With these values of p and q, factorise the given polynomial completely.

If (2x + 1) is a factor of both the expressions 2x2 - 5x + p and 2x2 + 5x + q, find the values of p and q. Hence, find the other factors of both the polynomials.

If a polynomial f(x) = x4 - 2x3 + 3x2 - ax - b leaves remainders 5 and 19 when divided by (x - 1) and (x + 1) respectively, find the values of a and b. Hence, determine the remainder when f(x) is divided by (x - 2).

If (x + 3) and (x - 4) are factors of x³ + ax² - bx + 24, find the values of a and b. With these values of a and b, factorise the given expression.

Use factor theorem to factorise the following polynomials completely :
(i) 4x³ + 4x² - 9x - 9
(ii) x³ - 19x - 30
(iii) 2x³ - x² - 13x - 6

If x³ - 2x² + px + q has a factor (x + 2) and leaves a remainder 9 when divided by (x + 1), find the values of p and q. With these values of p and q, factorise the given polynomial completely.

If (2x + 1) is a factor of both the expressions 2x² - 5x + p and 2x² + 5x + q, find the values of p and q. Hence, find the other factors of both the polynomials.

If a polynomial f(x) = x⁴ - 2x³ + 3x² - ax - b leaves remainders 5 and 19 when divided by (x - 1) and (x + 1) respectively, find the values of a and b. Hence, determine the remainder when f(x) is divided by (x - 2).