Without actual division, prove that x 4 + 2x 3 - 2x 2 + 2x - 3 is exactly divisible by x 2 + 2x - 3.

Let, f(x) = x 4 + 2x 3 - 2x 2 + 2x - 3 g(x) = x 2 + 2x - 3 = x 2 + 3x - x - 3 = x(x + 3) - 1(x + 3) = (x - 1)(x + 3) (x - 1) and (x + 3) are factors of g(x). In order to prove that f(x) is exactly divisible by g(x), it is sufficient to prove that x - 1 and x + 3 are factors of f(x) i.e. it is sufficient to show that f(1) = 0 and f(-3) = 0. Now, f(1) = (1) 4 + 2(1) 3 - 2(1) 2 + 2(1) - 3 = 1 + 2 - 2 + 2 - 3 = 0 f(-3) = (-3) 4 + 2(-3) 3 - 2(-3) 2 + 2(-3) - 3 = 81 - 54 - 18 - 6 - 3 = 0 ∴ f(x) is divisible by (x - 1) and (x + 3) Hence, f(x) is exactly divisible by g(x).

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

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By factor theorem, show that (x + 3) and (2x - 1) are the factors of 2x² + 5x - 3.

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Show that (x - 2) is a factor 3x² - x - 10. Hence, factorize 3x² - x - 10.

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Using factor theorem, show that (x - 2) is a factor of x³ + x² - 4x - 4. Hence, factorise the polynomial completely.

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Mathematics

Without actual division, prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.

Factorisation

Related Questions

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

By factor theorem, show that (x + 3) and (2x - 1) are the factors of 2x² + 5x - 3.

Show that (x - 2) is a factor 3x² - x - 10. Hence, factorize 3x² - x - 10.

Using factor theorem, show that (x - 2) is a factor of x³ + x² - 4x - 4. Hence, factorise the polynomial completely.

Mathematics

Without actual division, prove that x4 + 2x3 - 2x2 + 2x - 3 is exactly divisible by x2 + 2x - 3.

Factorisation

Related Questions

Using the remainder theorem, find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

By factor theorem, show that (x + 3) and (2x - 1) are the factors of 2x2 + 5x - 3.

Show that (x - 2) is a factor 3x2 - x - 10. Hence, factorize 3x2 - x - 10.

Using factor theorem, show that (x - 2) is a factor of x3 + x2 - 4x - 4. Hence, factorise the polynomial completely.

Without actual division, prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

By factor theorem, show that (x + 3) and (2x - 1) are the factors of 2x² + 5x - 3.

Show that (x - 2) is a factor 3x² - x - 10. Hence, factorize 3x² - x - 10.

Using factor theorem, show that (x - 2) is a factor of x³ + x² - 4x - 4. Hence, factorise the polynomial completely.