obtain all zeros of x^4-7x^3+17x^2-17x+6 if two of its zeros are 3 and 1
Do a division of x^4-7x^3+17x^2-17x+6 by (x-3)(x-1)=x²-4x+3 to get
x²-3x+2 as a quotient.
Factor x²-3x+2, equate to zero and solve for x.
To find the zeros of the polynomial x^4 - 7x^3 + 17x^2 - 17x + 6 given that two of its zeros are 3 and 1, we can make use of polynomial division or synthetic division.
Step 1: Divide the polynomial by (x - 3).
To do this, write the polynomial in descending order and perform the division:
1 | 1 -7 17 -17 6
- 1 - 6 11
1 - 6 11 - 6 0
The remainder is 0, which means (x - 3) is a factor of the polynomial.
Step 2: Divide the quotient obtained from step 1 by (x - 1).
Write the quotient in descending order and perform the division:
1 | 1 -6 11 -6
- 1 -5 -6
1 -5 6 0
The remainder is 0, which means (x - 1) is also a factor of the polynomial.
Step 3: Find the remaining zeros.
Now we have a quadratic equation, and we can use factoring or the quadratic formula to find the remaining zeros.
The quadratic equation is: x^2 - 5x + 6 = 0.
Factorization:
(x - 2)(x - 3) = 0
The remaining zeros are x = 2 and x = 3.
Hence, the zeros of the polynomial x^4 - 7x^3 + 17x^2 - 17x + 6, given that two of its zeros are 3 and 1, are 1, 2, 3, and 3.
To find all the zeros of the given polynomial, we can use polynomial division and factoring. Here's how you can obtain all the zeros:
Step 1: Since the polynomial has two known zeros, which are 3 and 1, we know that (x - 3) and (x - 1) are factors of the polynomial.
Step 2: Perform polynomial division by dividing the given polynomial by (x - 3) first.
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(x - 3) | x^4 - 7x^3 + 17x^2 - 17x + 6
To perform division, we start by dividing the highest degree term:
x^4 ÷ (x - 3) = x^3
Multiply the divisor by x^3 and subtract it from the dividend:
(x - 3)(x^3) = x^4 - 3x^3
Subtracting this from the dividend: x^4 - 7x^3 - (x^4 - 3x^3) = -4x^3
Bring down the next term: -4x^3 + 17x^2
Now repeat the process by dividing -4x^3 + 17x^2 by (x - 3):
-4x^3 ÷ (x - 3) = -4x^2
Multiply the divisor by -4x^2 and subtract it from the dividend:
(x - 3)(-4x^2) = -4x^3 + 12x^2
Subtracting this from the dividend: -4x^3 + 17x^2 - (-4x^3 + 12x^2) = 5x^2 + 17x^2 = 22x^2
Bring down the next term: 22x^2 - 17x
Repeat the process: divide 22x^2 - 17x by (x - 3):
22x^2 ÷ (x - 3) = 22x
Multiply the divisor by 22x and subtract it from the dividend:
(x - 3)(22x) = 22x^2 - 66x
Subtracting this from the dividend: 22x^2 - 17x - (22x^2 - 66x) = 49x
Bring down the next term: 49x - 6
Finally, divide 49x - 6 by (x - 3):
49x ÷ (x - 3) = 49
Multiply the divisor by 49 and subtract it from the dividend:
(x - 3)(49) = 49x - 147
Subtracting this from the dividend: 49x - 6 - (49x - 147) = 141
Now, we have the remainder 141. If the remainder is 0, it means (x - 3) is a factor of the polynomial, but in this case, it is not. So, (x - 3) is not a zero of the polynomial.
Step 3: Next, perform polynomial division by dividing the polynomial by (x - 1).
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(x - 1) | x^4 - 7x^3 + 17x^2 - 17x + 6
Follow the same steps as before to perform polynomial division.
(x - 1)(x^3) = x^4 - x^3
(x - 1)(-x^3) = -x^4 + x^3 = -8x^3
Subtract: -8x^3 + 17x^2 - 17x + 6 - (-8x^3) = 17x^2 - 17x + 6
Repeat the process with 17x^2 - 17x + 6:
(x - 1)(17x^2) = 17x^3 - 17x^2
(x - 1)(-17x^2) = -17x^3 + 17x^2 = 34x^2
Subtract: 34x^2 - 17x + 6 - (34x^2) = -17x + 6
Repeat again:
(x - 1)(-17x) = -17x^2 + 17x
(x - 1)(17x) = 17x^2 - 17x = 0
Subtract: 0 - (-17x) = 17x
Bring down the next term: 17x + 6
Finally, divide 17x + 6 by (x - 1):
(x - 1)(17) = 17x - 17
Subtract: 17x + 6 - (17x - 17) = 23
Now, the remainder 23 is non-zero, which means (x - 1) is not a factor of the polynomial.
Step 4: Now we have factored the polynomial as much as possible. We have: x^4 - 7x^3 + 17x^2 - 17x + 6 = (x - 3)(x - 1)(f(x))
Where f(x) represents the remaining factor of the polynomial that cannot be factored further.
To find the additional zeros of the polynomial, we can set the remaining factor f(x) equal to zero and solve for x.
f(x) = 0
So, to obtain all the zeros of the polynomial, you would need to find the zeros of the f(x) using algebraic methods such as factoring, the quadratic formula, or synthetic division.