Given that, x^2-4x+4 is a root of
x^3+ax^2+bx-4.
Find a and b
please show workings
#thanks
To find the values of "a" and "b" in the equation x^3 + ax^2 + bx - 4, you can use the fact that x^2 - 4x + 4 is a root of this equation.
When a polynomial has a root, it means that when you substitute that root value into the polynomial, the polynomial equation will equal zero. Therefore, to find "a" and "b", we will substitute x^2 - 4x + 4 into x^3 + ax^2 + bx - 4 and set it equal to zero.
Substituting x^2 - 4x + 4 into x^3 + ax^2 + bx - 4, we have:
(x^3 + ax^2 + bx - 4) = 0
[(x^2 - 4x + 4)^3 + a(x^2 - 4x + 4)^2 + b(x^2 - 4x + 4) - 4] = 0
Expanding each term, we have:
[(x^2 - 4x + 4)(x^2 - 4x + 4)(x^2 - 4x + 4) + a(x^2 - 4x + 4)(x^2 - 4x + 4) + b(x^2 - 4x + 4) - 4] = 0
Simplifying, we get:
(x^6 - 12x^5 + 54x^4 - 88x^3 + 45x^2 + 4x - 48) + a(x^4 - 8x^3 + 24x^2 - 32x + 16) + b(x^2 - 4x +4) - 4 = 0
Combining like terms, we have:
x^6 - 12x^5 + 54x^4 - 88x^3 + (45 + a)x^2 + (4 - 96 + 72a + b)x - (48 + 16a - 4b) = 0
Since x^2 - 4x + 4 is a root of the equation, it means that when we substitute it into the equation, the equation will equal zero. Therefore, we can set the coefficients of corresponding powers of x equal to zero.
Comparing the corresponding coefficients, we get:
Coefficient of x^6: 1 = 0
Coefficient of x^5: -12 = 0
Coefficient of x^4: 54 = 0
Coefficient of x^3: -88 = 0
Coefficient of x^2: 45 + a = 0 (Equation 1)
Coefficient of x: 4 - 96 + 72a + b = 0 (Equation 2)
Constant term: -48 + 16a - 4b = 0 (Equation 3)
Solving Equations 1, 2, and 3 simultaneously will give us the values of "a" and "b".
From Equation 1, we have a = -45.
Substituting the value of a in Equation 2, we get b = 4 - 96 + 72(-45) = -3212.
Therefore, a = -45 and b = -3212.
So, the values of "a" and "b" are -45 and -3212, respectively.