this is the third and final problem I can not figure out. Help please
Divide
(x^4 - 8x^2 + 15) / (x^2 -3) =
The numerator can be factored:
(x^4 - 8x^2 + 15) = (x^2 - 3)(x^2 - 5)
Use that and cancel the common (x^2 - 3) in the numerator and denominator.
(x^4 - 8x^2 + 15) = (x^2-3)(x^2-5)
so there will be a (x^2-3) term in the numerator and denominator that will cancel leaving (x^2-5)
on bady
To divide the polynomial (x^4 - 8x^2 + 15) by (x^2 - 3), we can use polynomial long division. Here's how you can do it step by step:
Note: I'll use "^" to represent the exponent notation.
Step 1: Start by dividing the highest power terms of the dividend and the divisor. In this case, divide x^4 by x^2, which gives you x^2.
______________________
x^2 - 3 | x^4 - 8x^2 + 15
x^2
Step 2: Multiply the divisor, (x^2 - 3), by the quotient you just found, x^2. This gives you x^4 - 3x^2.
______________________
x^2 - 3 | x^4 - 8x^2 + 15
- (x^4 - 3x^2)
x^2 - 3x^2
Step 3: Subtract the result obtained in step 2 from the dividend. In this case, subtract (x^2 - 3x^2) from (x^4 - 8x^2), which gives you -5x^2.
______________________
x^2 - 3 | x^4 - 8x^2 + 15
- (x^4 - 3x^2)
______________
-5x^2 + 15
Step 4: Bring down the next term from the dividend, which is +15 in this case.
______________________
x^2 - 3 | x^4 - 8x^2 + 15
- (x^4 - 3x^2)
______________
-5x^2 + 15
15
Step 5: Now, repeat steps 1 to 4 using the new dividend (-5x^2 + 15) to divide it by the divisor (x^2 - 3).
Step 1: Divide -5x^2 by x^2, which gives you -5.
______________________
x^2 - 3 | x^4 - 8x^2 + 15
- (x^4 - 3x^2)
______________
-5x^2 + 15
- (-5x^2)
-5x^2 + 15
Step 2: Multiply the divisor, (x^2 - 3), by the quotient you just found, -5. This gives you -5x^2 + 15.
______________________
x^2 - 3 | x^4 - 8x^2 + 15
- (x^4 - 3x^2)
______________
-5x^2 + 15
- (-5x^2 + 15)
_________________
0
Step 3: Subtract the result obtained in step 2 from the dividend. In this case, subtract (-5x^2 + 15) from (-5x^2 + 15), which gives you 0.
Since the resulting polynomial is zero, we can conclude that the quotient is x^2 - 5, and there is no remainder.
Therefore, the answer is: (x^4 - 8x^2 + 15) / (x^2 - 3) = x^2 - 5.