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प्रश्न
Using the Reminder Theorem, factorise of the following completely.
2x3 + x2 – 13x + 6
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उत्तर
Let f (x) = 2x3 + x2 − 13x + 6
For x = 2,
f(x) = f(2) = 2(2)3 + (2)2 − 13(2) + 6 = 16 + 4 − 26 + 6 = 0
Hence, (x − 2) is a factor of f(x).
∴ `2x^3 + x^2 - 13x + 6 = (x-2) (2x^2 + 5x -3)`
= `(x-2)(2x^2 + 6x -x -3)`
= `(x-2) [2x(x + 3) -(x-3)]`
=` (x - 2)(x + 3) (2x - 1)`
संबंधित प्रश्न
When divided by x – 3 the polynomials x3 – px2 + x + 6 and 2x3 – x2 – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’.
Using the Remainder Theorem, factorise each of the following completely.
3x3 + 2x2 – 23x – 30
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).
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.
If (x + 1) and (x – 2) are factors of x3 + (a + 1)x2 – (b – 2)x – 6, find the values of a and b. And then, factorise the given expression completely.
If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x2 - 3x + 5a
If (x – 2) is a factor of 2x3 – x2 + px – 2, then
(i) find the value of p.
(ii) with this value of p, factorise the above expression completely
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.
f 2x3 + ax2 – 11x + b leaves remainder 0 and 42 when divided by (x – 2) and (x – 3) respectively, find the values of a and b. With these values of a and b, factorize the given expression.
If (2x + 1) is a factor of both the expressions 2x2 – 5x + p and 2x2 + 5x + q, find the value of p and q. Hence find the other factors of both the polynomials.
