Math Problem

Question

Determine the value(s) of b so that x+b is a factor of p(x)= x^3+(b-2)x^2+x+(12-b).

Answer 1

Determine the value(s) of b so that x+b is a factor of p(x)= x^3+(b-2)x^2+x+(12-b).

 

An alternative method to solve this problem avoids the use of synthetic or long division.

Use the Remainder- Factor Theorem. If you substitute x = -b, you would obtain

p(-b) = -2b² - 2b + 12 (show this).

This is the remainder  which must  equal zero if x + b is a factor.

Solve -2b² - 2b + 12 = 0 to obtain two values of b.

Answer 2

We can use synthetic division to find the values of b.  The coefficients of p(x) is the dividend.  The divisor is the root of the factor (x + b).  The divisor is then -b.

 

 

Setting up the division,

 

 

-b | 1          (b - 2)         1              (12 - b)

 

                    -b             2b            (-2b² - b)

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      1            -2          (2b + 1)             0

 

 

 

Allow me to explain the process.  I place the first coefficient below the line in the first column.  Then I multiply that digit by the divisor.  The product became the addend in the next column.  I then added the digits in the second column.  I multiply the sum in this column by the divisor.  The product became the addend in the succeeding column.  Then I repeated the process until I get to the last column.

 

Now, we use the last column to set up an equation to solve for b.  Notice that the last digit under the line in the division is zero.  This last digit is the remainder. 

 

12 - b - 2b² - b = 0

 

-2b² - 2b + 12 = 0

 

-2(b² + b - 6) = 0

 

-2(b + 3)(b - 2) = 0

 

b = -3        and        b = 2

 

 

Now just verify these values of b by plugging them into the division.  Either one of them or both of the are solutions.

Answer 3

For x + b to be a factor, p(x) / (x + b) must have a remainder of zero.

 

By synthetic division, p(x) / (x + b) has a remainder of -2b² - 2b + 12

 

-2b² - 2b + 12 = 0             multiply both sides by -1

2b² + 2b - 12 = 0              divide both side by 2

b² + b - 6 = 0                    FOIL

(b + 3)(b - 2) = 0              product rule

{-3, 2}