Find each solution using long division. Show your work.
Show that x + 1 is NOT a factor of...
2x^3 + x^2 - 11x - 30.
Show that x - 3 IS a factor of 2x^3 - 11x^2 - 11x^2 + 12x + 9.
Just do long division like you would for anything else. Look at the remainder in the quotient after you divide the divisor into the dividend.
P(-1) -2 +1 +11 +30
P(-1) =40
P(-1) Is not 0
so x+1 is not a factor the question.
P(3) 54 -99 -99 +36 + 9
P(3) =-99
x-3 is NOT a factor of the 2nd question.
To determine whether a polynomial is a factor of another polynomial, we can use long division. I will show the step-by-step process for each case.
1. To show that x + 1 is not a factor of 2x^3 + x^2 - 11x - 30:
Step 1: Begin by dividing the leading term of the dividend (2x^3) by the leading term of the divisor (x + 1). This gives us 2x^2 as the quotient.
2x^2
--------------
x + 1 | 2x^3 + x^2 - 11x - 30
Step 2: Multiply the divisor (x + 1) by the quotient (2x^2) and subtract the result from the dividend.
2x^2
--------------
x + 1 | 2x^3 + x^2 - 11x - 30
- (2x^3 + 2x^2)
-x^2 - 11x - 30
Step 3: Repeat the process by dividing the new dividend (-x^2 - 11x - 30) by the divisor (x + 1).
2x^2 - x
--------------
x + 1 | 2x^3 + x^2 - 11x - 30
- (2x^3 + 2x^2)
--------------
-x^2 - 11x - 30
- (-x^2 - x)
-10x - 30
Step 4: Divide the new dividend (-10x - 30) by the divisor (x + 1).
2x^2 - x - 10
--------------
x + 1 | 2x^3 + x^2 - 11x - 30
- (2x^3 + 2x^2)
--------------
-x^2 - 11x - 30
- (-x^2 - x)
--------------
-10x - 30
- (-10x - 10)
--------------
-20
The remainder is -20, which is non-zero. Therefore, x + 1 is not a factor of 2x^3 + x^2 - 11x - 30.
2. To show that x - 3 is a factor of 2x^3 - 11x^2 - 11x^2 + 12x + 9:
Step 1: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x - 3). This gives us 2x^2 as the quotient.
2x^2
--------------
x - 3 | 2x^3 - 11x^2 - 11x^2 + 12x + 9
Step 2: Multiply the divisor (x - 3) by the quotient (2x^2) and subtract the result from the dividend.
2x^2
--------------
x - 3 | 2x^3 - 11x^2 - 11x^2 + 12x + 9
- (2x^3 - 6x^2)
-5x^2 + 12x + 9
Step 3: Repeat the process by dividing the new dividend (-5x^2 + 12x + 9) by the divisor (x - 3).
2x^2 - 5x
--------------
x - 3 | 2x^3 - 11x^2 - 11x^2 + 12x + 9
- (2x^3 - 6x^2)
--------------
-5x^2 + 12x + 9
- (-5x^2 + 15x)
-3x + 9
Step 4: Divide the new dividend (-3x + 9) by the divisor (x - 3).
2x^2 - 5x + 3
--------------
x - 3 | 2x^3 - 11x^2 - 11x^2 + 12x + 9
- (2x^3 - 6x^2)
--------------
-5x^2 + 12x + 9
- (-5x^2 + 15x)
--------------
-3x + 9
- (-3x + 9)
0
The remainder is 0, which means x - 3 is a factor of 2x^3 - 11x^2 - 11x^2 + 12x + 9.
I have shown the step-by-step long division process to demonstrate whether x + 1 is a factor or not and whether x - 3 is a factor or not.