When 4x^3 - 3ax + 5 is subtracted from 11x^3 + ax^2 - x + b, the result is cx^3 - 2x^2 + dx - 1.
How do you find a, b, c, and d?
To find the values of a, b, c, and d, we need to compare the coefficients of corresponding terms on both sides of the equation.
The given equation is:
(11x^3 + ax^2 - x + b) - (4x^3 - 3ax + 5) = cx^3 - 2x^2 + dx - 1
Step 1: Simplify the equation by subtracting the polynomials on both sides.
11x^3 + ax^2 - x + b - 4x^3 + 3ax - 5 = cx^3 - 2x^2 + dx - 1
Simplifying further, we get:
7x^3 + ax^2 + (3a - 1)x + (b - 5) = cx^3 - 2x^2 + dx - 1
Step 2: Now, compare the coefficients of corresponding terms on both sides of the equation.
For the x^3 term:
7 = c
For the x^2 term:
a = -2
For the x term:
3a - 1 = d
=> substituting the value of a, we have
3(-2) - 1 = d
=> -6 - 1 = d
=> d = -7
For the constant term:
b - 5 = -1
=> b = 4
Therefore, the values of a, b, c, and d are:
a = -2
b = 4
c = 7
d = -7
To find the values of a, b, c, and d, we can equate the coefficients of corresponding powers of x on both sides of the equation.
Given:
(11x^3 + ax^2 - x + b) - (4x^3 - 3ax + 5) = (cx^3 - 2x^2 + dx - 1)
Expanding the equation, we get:
11x^3 + ax^2 - x + b - 4x^3 + 3ax - 5 = cx^3 - 2x^2 + dx - 1
Now, let's group like terms:
x^3 terms:
11x^3 - 4x^3 = cx^3
7x^3 = cx^3
Comparing the coefficients, we find that c = 7.
x^2 terms:
ax^2 - 2x^2 = -2x^2
(a - 2)x^2 = -2x^2
Comparing the coefficients, we find that a - 2 = -2, so a = 0.
x terms:
-x + 3ax = dx
(3a - 1)x = dx
Comparing the coefficients, we find that 3a - 1 = d, so d = 3(0) - 1 = -1.
Constant terms:
b - 5 = -1
b = 4
Therefore, the values of a, b, c, and d are:
a = 0
b = 4
c = 7
d = -1
(11x^3 + ax^2 - x + b)-(4x^3 - 3ax + 5) = 7x^3 + ax^2 - (3a+1)x + (b+5)
That means
c = 7
a = 2
d = -(3a+1) = -7
b+5 = -1, so b = -6