Find all the zeroes of the polynomial
2x^4+7x^3-19x^2-14x+30 , if two of its zeroes are root2 and -root2?
(x-√2)(x+√2) = x^2-2
(2x^4+7x^3-19x^2-14x+30)/(x^2-2) = 2x^2+7x-15
Now you can factor that to find the other two roots
To find the zeroes of the polynomial, we will use the fact that if a value is a zero of the polynomial, then the polynomial evaluated at that value will be equal to zero.
Given that the polynomial has two zeroes: √2 and -√2, we can write two equations:
1. 2(√2)^4 + 7(√2)^3 - 19(√2)^2 - 14(√2) + 30 = 0
2. 2(-√2)^4 + 7(-√2)^3 - 19(-√2)^2 - 14(-√2) + 30 = 0
Simplifying these equations gives us:
1. 2(2) + 7(2√2) - 19(2) - 14√2 + 30 = 0
4 + 14√2 - 38 - 14√2 + 30 = 0
-4 + 14√2 = 0
2. 2(2) + 7(-2√2) - 19(2) + 14√2 + 30 = 0
4 - 14√2 - 38 + 14√2 + 30 = 0
-4 + 14√2 = 0
In both cases, we get the equation -4 + 14√2 = 0.
To solve this equation, we can add 4 to both sides:
14√2 = 4
Then, divide both sides by 14:
√2 = 4/14
Simplifying the right side gives us:
√2 = 2/7
Since this is not true (√2 ≠ 2/7), we have reached a contradiction.
Thus, there are no other zeroes of the polynomial other than √2 and -√2.
To find the zeros of the polynomial, let's begin by factoring out the given roots, root2 and -root2, from the polynomial.
First, let's set up the factors corresponding to the given roots:
(x - root2) and (x + root2)
To find the other factors, we can use long division or synthetic division method.
Using synthetic division:
Step 1: Set up the synthetic division table:
_________________________________
-root2 | 2 7 -19 -14 30
_________________________
Step 2: Bring down the coefficients of the polynomial into the synthetic division table:
_________________________________
-root2 | 2 7 -19 -14 30
_________________________
Step 3: Multiply the divisor (-root2) by the first term (2) and write the result under the second term (7):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
_________________________
Step 4: Add the second term (7) and the result obtained in the previous step (-4root2):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
_________________________
3 3
Step 5: Multiply the divisor (-root2) by the current term (3) and write the result under the third term (-19):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
_________________________
3 3
6root2
_________________________
Step 6: Add the third term (-19) and the result obtained in the previous step (6root2):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
3 3
6root2
_________________________
3 3 -19
Step 7: Multiply the divisor (-root2) by the current term (3) and write the result under the fourth term (-14):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
3 3
6root2
_________________________
3 3 -19
-6root2
_________________________
Step 8: Add the fourth term (-14) and the result obtained in the previous step (-6root2):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
3 3
6root2
_________________________
3 3 -19
-6root2
-3
Step 9: Multiply the divisor (-root2) by the current term (3) and write the result under the fifth term (30):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
3 3
6root2
_________________________
3 3 -19
-6root2
-3
_________________________
Step 10: Add the fifth term (30) and the result obtained in the previous step (-3):
_________________________________
-root2 | 2 7 -19 -14 30
-4root2
3 3
6root2
_________________________
3 3 -19
-6root2
-3
-3root2
Step 11: Now we have the remainder (-3root2). Since the remainder is not zero, -root2 is not a zero of the polynomial.
Now, let's divide the polynomial by (x - root2) and (x + root2) using long division to find the other roots.
(2x^4 + 7x^3 - 19x^2 - 14x + 30)/(x - root2)/(x + root2)
The long division process can be complex to explain through text, but you can do it by dividing the polynomial step by step. After performing the long division, you will get a quadratic equation. Solving that equation will give you the other two zeros of the polynomial.