Advertisements
Advertisements
प्रश्न
Factorise the expression f(x) = 2x3 – 7x2 – 3x + 18. Hence, find all possible values of x for which f(x) = 0.
Advertisements
उत्तर
f(x) = 2x3 – 7x2 – 3x + 18
For x = 2,
f(x) = f(2)
= 2(2)3 – 7(2)2 – 3(2) + 18
= 16 – 28 – 6 + 18
= 0
Hence, (x – 2) is a factor of f(x).
2x2 – 3x – 9
`x - 2")"overline(2x^3 - 7x^2 - 3x + 18)`
2x3 – 4x2
– +
– 3x2 – 3x
– 3x2 + 6x
+ –
– 9x + 18
– 9x + 18
+ –
0
∴ 2x3 – 7x2 – 3x + 18 = (x – 2)(2x2 – 3x – 9)
= (x – 2)(2x2 – 6x + 3x – 9)
= (x – 2)[2x(x – 3) + 3(x – 3)]
= (x – 2)(x – 3)(2x + 3)
Now, f(x) = 0
⇒ 2x3 – 7x2 – 3x + 18 = 0
⇒ (x – 2)(x – 3)(2x + 3) = 0
⇒ `x = 2, 3, (-3)/2`
संबंधित प्रश्न
A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.
Find the number that must be subtracted from the polynomial 3y3 + y2 – 22y + 15, so that the resulting polynomial is completely divisible by y + 3.
The polynomial px3 + 4x2 – 3x + q is completely divisible by x2 – 1; find the values of p and q. Also, for these values of p and q, factorize the given polynomial completely.
In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:
x3 + ax2 − 2x + a + 4
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 x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x5 - 3x4 - ax3 + 3ax2 + 2ax + 4.
Show that x2 - 9 is factor of x3 + 5x2 - 9x - 45.
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 value of p and q. Hence find the other factors of both the polynomials.
While factorizing a given polynomial using the remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x3 + 7x2 + 2x – 3.
- Is the student’s solution correct in stating that (2x + 1) is a factor of the given polynomial?
- Give a valid reason for your answer.
Also, factorize the given polynomial completely.
