If 4x^3-6x^2+1=(x-2)Q(x)+ax-3 for all values of x ,where Q(x) is a polynomial in x ,find
(i)the value of constant a
(ii)Q(x)
the statement must be true for all values of x
suppose we let x = 2 , (can you see why ?)
then
4x^3-6x^2+1=(x-2)Q(x)+ax-3
32 - 24 + 1 = (0)(Q(x) + 2a - 3
9 = 2a - 3
2a = 12
a = 6
so 4x^3-6x^2+1=(x-2)Q(x)+ax-3
becomes
4x^3-6x^2+1=(x-2)Q(x)+6x-3
4x^3 - 6x^2 - 6x + 3 = (x-2)Q(x)
Q(x) = (4x^3 - 6x^2 - 6x + 3)/(x-2)
I noticed your other post is just about the same type.
(i) To find the value of constant a, we can use the remainder theorem. According to the theorem, when we divide the polynomial 4x^3 - 6x^2 + 1 by (x - 2), the remainder should be equal to ax - 3.
Let's perform the long division:
4x^2 + 2x - 1
______________________
(x - 2) | 4x^3 - 6x^2 + 0x + 1
By dividing 4x^3 - 6x^2 + 0x + 1 by (x - 2), we obtain a quotient of 4x^2 + 2x - 1.
The remainder is 0 (no remainder). Therefore, the equation becomes:
0 = (x - 2)(4x^2 + 2x - 1) + ax - 3
Since the remainder is 0, the coefficient of the x term in the remainder (ax - 3) must also be zero. Therefore, a = 0.
(ii) To find Q(x), we just need to divide the given polynomial (4x^3 - 6x^2 + 1) by (x - 2) using long division. The quotient we obtain will be Q(x).
Performing long division:
4x^2 + 2x - 1
______________________
(x - 2) | 4x^3 - 6x^2 + 0x + 1
By dividing 4x^3 - 6x^2 + 0x + 1 by (x - 2), we obtain a quotient of 4x^2 + 2x - 1.
Therefore, Q(x) = 4x^2 + 2x - 1.
To find the value of constant "a" and the polynomial "Q(x)", we will use polynomial division.
Step 1: Set up the polynomial division by dividing 4x^3 - 6x^2 + 1 by (x - 2).
_______________________
(x - 2) | 4x^3 - 6x^2 + 0x + 1
Step 2: Divide the first term of the dividend (4x^3) by the first term of the divisor (x) to get 4x^2.
Step 3: Multiply the divisor (x - 2) by the quotient from step 2 (4x^2) to get 4x^3 - 8x^2.
_______________________
(x - 2) | 4x^3 - 6x^2 + 0x + 1
- (4x^3 - 8x^2)
_________________
2x^2 + 0x
Step 4: Bring down the next term from the dividend (0x) and divide it by the first term of the divisor (x) to get 0.
Step 5: Multiply the divisor (x - 2) by the quotient from step 4 (0) to get 0x.
_______________________
(x - 2) | 4x^3 - 6x^2 + 0x + 1
- (4x^3 - 8x^2)
_________________
2x^2 + 0x
- (0)
________
2x^2 + 0x
Step 6: Bring down the last term from the dividend (1) and divide it by the first term of the divisor (x) to get 1.
Step 7: Multiply the divisor (x - 2) by the quotient from step 6 (1) to get x - 2.
_______________________
(x - 2) | 4x^3 - 6x^2 + 0x + 1
- (4x^3 - 8x^2)
_________________
2x^2 + 0x
- (0)
________
2x^2 + 0x
- (2x - 4)
_________
4x - 3
Step 8: The remainder of the polynomial division is 4x - 3.
So, we have: 4x^3 - 6x^2 + 1 = (x - 2)Q(x) + (4x - 3)
Comparing this equation with the original equation: 4x^3 - 6x^2 + 1 = (x - 2)Q(x) + ax - 3
(i) From the comparison, we can determine that a = 4.
(ii) The polynomial Q(x) is the quotient we obtained: Q(x) = 2x^2 + 0x - (2x - 4) = 2x^2 - 2x + 4
To find the value of constant "a" and the polynomial "Q(x)" that satisfies the equation 4x^3-6x^2+1 = (x-2)Q(x) + ax-3 for all values of x, we need to use polynomial division.
(i) Finding the value of constant "a":
Step 1: We divide the left-hand side of the equation 4x^3-6x^2+1 by the divisor (x-2):
____________________
(x-2) | 4x^3 - 6x^2 + 1
Step 2: Divide the first term of the dividend (4x^3) by the first term of the divisor (x), which gives 4x^2.
Step 3: Multiply the result from the previous step (4x^2) by the divisor (x-2), which gives 4x^3 - 8x^2.
Step 4: Subtract the result obtained in the previous step (4x^3 - 8x^2) from the original dividend (4x^3 - 6x^2 + 1). The subtraction gives 2x^2 + 1.
____________________
(x-2) | 4x^3 - 6x^2 + 1
-(4x^3 - 8x^2)
________________
2x^2 + 1
Step 5: Divide the first term of the new dividend (2x^2) by the first term of the divisor (x), which gives 2x.
Step 6: Multiply the result from the previous step (2x) by the divisor (x-2), which gives 2x^2 - 4x.
Step 7: Subtract the result obtained in the previous step (2x^2 - 4x) from the new dividend (2x^2 + 1). The subtraction gives 4x + 1.
____________________
(x-2) | 4x^3 - 6x^2 + 1
-(4x^3 - 8x^2)
________________
2x^2 + 1
-(2x^2 - 4x)
______________
4x + 1
Step 8: Divide the first term of the new dividend (4x) by the first term of the divisor (x), which gives 4.
Step 9: Multiply the result from the previous step (4) by the divisor (x-2), which gives 4x - 8.
Step 10: Subtract the result obtained in the previous step (4x - 8) from the new dividend (4x + 1). The subtraction gives 9.
____________________
(x-2) | 4x^3 - 6x^2 + 1
-(4x^3 - 8x^2)
________________
2x^2 + 1
-(2x^2 - 4x)
______________
4x + 1
-(4x - 8)
______________
9
Step 11: Now, we have a remainder of 9. Since this remainder must be equal to the expression ax-3 in the original equation, we can equate the terms to determine the value of constant "a":
9 = ax - 3
From this equation, we can see that a = 9.
(ii) Finding the polynomial Q(x):
After performing polynomial division, we have the quotient 2x^2 + 2x + 4 and the remainder 9. Therefore, the polynomial Q(x) is 2x^2 + 2x + 4.
To summarize:
(i) The value of constant "a" is 9.
(ii) The polynomial Q(x) is 2x^2 + 2x + 4.