If a polynomial f(x) = x 4 - 2x 3 + 3x 2 - 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).

By remainder theorem, on dividing f(x) by (x - a), the remainder left is f(a). f(x) = x 4 - 2x 3 + 3x 2 - ax - b ∴ On dividing f(x) by (x + 1) or (x - (-1)), Remainder = f(-1) Given, on dividing by (x + 1) remainder = 19, ∴ f(-1) = 19 ∴ On dividing f(x) by (x - 1), Remainder = f(1) Given, on dividing by (x - 1) remainder = 5, ∴ f(1) = 5 Putting value of a = 13 + b from equation 1, Putting a = 5 and b = -8 in f(x) we get, f(x) = x 4 - 2x 3 + 3x 2 - 5x + 8. On dividing f(x) by (x - 2), remainder = f(2) by remainder theorem Hence, the value of a is 5 and b is -8. On dividing x 4 - 2x 3 + 3x 2 - 5x + 8 by (x - 2) the value of remainder is 10.

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.

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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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When a polynomial f(x) is divided by (x - 1), the remainder is 5 and when it is, divided by (x - 2), the remainder is 7. Find the remainder when it is divided by (x - 1)(x - 2).

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The polynomial 3x³ + 8x² - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

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Mathematics

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).

Factorisation

Related Questions

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.

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.

When a polynomial f(x) is divided by (x - 1), the remainder is 5 and when it is, divided by (x - 2), the remainder is 7. Find the remainder when it is divided by (x - 1)(x - 2).

The polynomial 3x³ + 8x² - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

Mathematics

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).

Factorisation

Related Questions

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.

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.

When a polynomial f(x) is divided by (x - 1), the remainder is 5 and when it is, divided by (x - 2), the remainder is 7. Find the remainder when it is divided by (x - 1)(x - 2).

The polynomial 3x3 + 8x2 - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

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.

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.

The polynomial 3x³ + 8x² - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.