Advertisements
Advertisements
प्रश्न
The polynomial 3x3 + 8x2 – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.
Advertisements
उत्तर
Given, P(x) = 3x3 + 8x2 – 15x + k
Put x – 1 = 0
x = 1
Now, P(1) = 3(1)3 + 8(1)2 – 15(1) + k = 0
⇒ 3 + 8 – 15 + k = 0
⇒ – 4 + k = 0
⇒ k = 4
Hence, k = 4
P(x) = 3x3 + 8x2 – 15x + 4
`x - 1")"overline(3x^3 + 8x^2 - 15x + 4)(3x^2 + 11x - 4`
3x3 – 3x2
– +
11x2 – 15x
11x2 – 11x
– +
– 4x + 4
– 4x + 4
+ –
x
∴ 3x3 + 8x2 – 15x + 4 = (x – 1) (3x2 + 11x – 4)
= (x – 1) (3x2 + 12x – x – 4)
= (x – 1) [3x(x + 4) – 1(x + 4)]
= (x – 1) (3x – 1) (x + 4)
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’.
Factorise the expression f(x) = 2x3 – 7x2 – 3x + 18. Hence, find all possible values of x for which f(x) = 0.
Show that (x – 1) is a factor of x3 – 7x2 + 14x – 8. Hence, completely factorise the given expression.
Using the factor theorem, show that (x - 2) is a factor of `x^3 + x^2 -4x -4 .`
Hence factorise the polynomial completely.
In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:
x6 - ax5 + x4 - ax3 + 3a + 2
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, factorize the given polynomial 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.
Factorize completely using factor theorem:
2x3 – x2 – 13x – 6
(x – 2) is a factor of ______.
