If (x − 2) is a factor of 5x2 − bx − 6, find the value of b. Show all of your work and explain your reasoning
If (x - 2) is a factor of 5x^2 - bx - 6, it means that if we substitute x = 2 into the expression 5x^2 - bx - 6, the expression should equal to zero.
So, let's evaluate the expression at x = 2:
5(2)^2 - b(2) - 6 = 0
20 - 2b - 6 = 0
14 - 2b = 0
-2b = -14
b = (-14)/(-2)
b = 7
Therefore, the value of b is 7.
To find the value of b, we can use the factor theorem. According to the factor theorem, if (x - 2) is a factor of the polynomial 5x^2 - bx - 6, then substituting x = 2 into the polynomial should give us an answer of 0.
Let's substitute x = 2 into the polynomial:
5(2)^2 - b(2) - 6
Simplifying this expression, we get:
20 - 2b - 6
Combining like terms, we have:
14 - 2b
Since we want this expression to equal zero, we can set it equal to zero and solve for b:
14 - 2b = 0
Subtracting 14 from both sides, we have:
-2b = -14
Dividing both sides by -2, we get:
b = 7
Therefore, the value of b is 7.
To determine the value of b, we need to use the fact that (x - 2) is a factor of 5x^2 - bx - 6.
When a polynomial P(x) is divided by a linear factor (x - a), the remainder is equal to P(a).
In our case, when we divide 5x^2 - bx - 6 by (x - 2), the remainder will be zero because (x - 2) is a factor. So, we have:
(5x^2 - bx - 6) / (x - 2) = 0
To simplify the division, we can use polynomial long division or synthetic division. Let's use synthetic division:
2 |
--------
5 | 0 - b - 6
-10| 10b 20
--------
10b - 10
Since the remainder is zero, we have 10b - 10 = 0.
To solve for b, we can isolate the variable:
10b - 10 = 0
Add 10 to both sides:
10b = 10
Divide both sides by 10:
b = 10/10
Simplifying it, we get:
b = 1
Therefore, the value of b is 1.