Is the binomial, x + 4 a factor of, x^3 + 7x^2 + 8x - 15? Show your work.
To determine if x + 4 is a factor of x^3 + 7x^2 + 8x - 15, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial P(x) is divided by (x - a), the remainder is equal to P(a).
In this case, we want to check if the remainder is zero when x + 4 is divided into x^3 + 7x^2 + 8x - 15.
We substitute -4 into the polynomial:
(-4)^3 + 7(-4)^2 + 8(-4) - 15
-64 + 7(16) - 32 - 15
-64 + 112 - 32 - 15
-64 + 112 - 47
= 1
The remainder is 1, which means that x + 4 is not a factor of x^3 + 7x^2 + 8x - 15.
To determine if the binomial x + 4 is a factor of the polynomial x^3 + 7x^2 + 8x - 15, we need to perform polynomial long division.
Step 1: Set up the division
Write the polynomial division like this:
________
x + 4 | x^3 + 7x^2 + 8x - 15
Step 2: Divide the leading term
Divide x^3 by x to get x^2. Write the result above the line.
x^2
x + 4 | x^3 + 7x^2 + 8x - 15
Step 3: Multiply the divisor by the quotient
Multiply the divisor x + 4 by the quotient x^2 and write the result below the line.
x^2
x + 4 | x^3 + 7x^2 + 8x - 15
- (x^3 + 4x^2)
Step 4: Subtract
Subtract the result from step 3 from the dividend:
x^2
x + 4 | x^3 + 7x^2 + 8x - 15
- (x^3 + 4x^2)
________________
3x^2 + 8x
Step 5: Bring down the next term
Bring down the next term from the dividend, which is 8x.
x^2 + (3x + 4)
x + 4 | x^3 + 7x^2 + 8x - 15
- (x^3 + 4x^2)
________________
3x^2 + 8x
- (3x^2 + 8x)
Step 6: Subtract
Subtract the result from step 5 from the remaining polynomial:
x^2 + (3x + 4)
x + 4 | x^3 + 7x^2 + 8x - 15
- (x^3 + 4x^2)
________________
3x^2 + 8x - (3x^2 + 8x)
___________
0
Step 7: Remainder
The result of the subtraction is 0, indicating that there is no remainder.
Since the remainder is 0, we can conclude that x + 4 is indeed a factor of the polynomial x^3 + 7x^2 + 8x - 15.
To determine whether the binomial x + 4 is a factor of the polynomial x^3 + 7x^2 + 8x - 15, we can use synthetic division to divide the polynomial by the binomial.
First, we set up the synthetic division by writing down the coefficients of the polynomial in descending order, along with the divisor:
-4 │ 1 7 8 -15
Next, we bring down the first coefficient, which is 1, and divide it by the divisor:
-4 │ 1 7 8 -15
-4
Then, we multiply the divisor, -4, by the result above (-4) and write it under the next coefficient:
-4 │ 1 7 8 -15
-4
____________
1 3 4
Next, we add the next coefficient, 3, to the result above (-4 + 3 = -1) and write it down:
-4 │ 1 7 8 -15
-4 -3
____________
1 3 4
-4
We repeat the process with the new result (-1) multiplied by the divisor (-4):
-4 │ 1 7 8 -15
-4 -3
____________
1 3 4
-4
____________
1 -1 0
Finally, we have the quotient of 1 - 1x + 0x^2, which represents the remaining terms after dividing by x + 4. The last term is zero, indicating that x + 4 is a factor of the polynomial x^3 + 7x^2 + 8x - 15.
Therefore, we can conclude that the binomial x + 4 is indeed a factor of the polynomial x^3 + 7x^2 + 8x - 15.