Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let p(x) = x5 - 3x4 - ax3 + 3ax2 + 2ax + 4 ...(i)
Since, (x - 2) is facxtor of p(x), so p(2) = 0
Put x = 2 in equation (i) we get
p(2) = (2)5 - 3(2)4 -a(2)3 + 3a(2)2 + 2a(2) + 4
= 32 - 3 x 16 - a x 8 + 3a x 4 + 4a + 4
= 32 - 48 - 8a + 12a + 4a + 4
= 8a - 12
But p(2) = 0
⇒ 8a = 12 = 0
⇒ a = `(12)/(8)`
⇒ a = `(3)/(2)`.
RELATED QUESTIONS
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.
Show that (x – 1) is a factor of x3 – 7x2 + 14x – 8. Hence, completely factorise the given expression.
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.
Using remainder Theorem, factorise:
2x3 + 7x2 − 8x – 28 Completely
Using the Reminder Theorem, factorise of the following completely.
2x3 + x2 – 13x + 6
Using the factor theorem, show that (x - 2) is a factor of `x^3 + x^2 -4x -4 .`
Hence factorise the polynomial completely.
Prove that (5x + 4) is a factor of 5x3 + 4x2 – 5x – 4. Hence 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.
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.
For the polynomial x5 – x4 + x3 – 8x2 + 6x + 15, the maximum number of linear factors is ______.
