Given that 2x-1 is a factor of x^3-3x^2+bx-b, use full sentences to describe the steps and the theorem you would use to determine the value of b algebraically.
the Remainder Theorem says that if x-a is factor of p(x), then p(a) = 0
so evaluate p(1/2) and evaluate b so the remainder is zero.
thanks
To determine the value of b algebraically, we can use the Factor Theorem, which states that if (x - a) is a factor of a polynomial, then plugging in the value of a into the polynomial will result in zero.
Step 1: We are given that 2x - 1 is a factor of x^3 - 3x^2 + bx - b.
Step 2: Applying the Factor Theorem, we set 2x - 1 equal to zero and solve for x, since it is a factor:
2x - 1 = 0
2x = 1
x = 1/2
Step 3: Substituting x = 1/2 into the polynomial x^3 - 3x^2 + bx - b and setting it equal to zero:
(1/2)^3 - 3(1/2)^2 + b(1/2) - b = 0
Step 4: Simplifying the equation:
1/8 - 3/4 + b/2 - b = 0
1/8 - 6/8 + b/2 - b = 0
-5/8 + b/2 - b = 0
Step 5: Combining like terms and isolating b:
b/2 - b = 5/8
(2b - 2b)/2 = 5/8
0 = 5/8
Step 6: From step 5, we can see that 0 is not equal to 5/8. So there is no value of b that satisfies the equation.
Hence, there is no value of b that would make 2x - 1 a factor of x^3 - 3x^2 + bx - b.