I am having touble understanding:
how to factor the polynomial :
8x^4 -6x^3 -57x^2 -6x-65, into its linear factors, given that i is a zero.
I tried to use synthetic division, but i keep getting a remainder... please help me understand how to solve it.
you are given that x = i is a solution
Complex roots, just like irrational roots, always come in conjugate pairs, so x = -i is another solution
so you know
(x+i) and (x-i) are factors
(x+1)(x-1) = x^2 + 1
which means x^2 + 1 is factor of your given polynomial
now do a long division of 8x^4 -6x^3 -57x^2 -6x-65 by (x^2 + 1)
I am not going to show this here, but you should get
8x^4 -6x^3 -57x^2 -6x-65
= (x^1 + 1)(8x^2 - 6x - 65)
I trust you will be able to further factor the last part, it does factor
To factor the given polynomial, 8x^4 - 6x^3 - 57x^2 - 6x - 65, into its linear factors, given that i is a zero, we can use synthetic division. However, we need to make sure that we're using the correct method.
When working with complex zeros like i (the imaginary unit), we need to remember that complex zeros occur in conjugate pairs. This means that if i is a zero, then its conjugate, -i, is also a zero.
Now, let's go through step-by-step on how to solve this:
Step 1: Write down the coefficients of the polynomial.
The coefficients for our polynomial, in descending order, are: 8, -6, -57, -6, -65.
Step 2: Set up synthetic division.
Using synthetic division, we set up our division like this:
i | 8 -6 -57 -6 -65
Step 3: Divide each term by i.
To divide each term by i, we can consider i as a complex number and use its properties. The general rule is that i^2 = -1.
Let's start by dividing the first term, 8, by i. To eliminate i, we'll multiply both the numerator and denominator by -i:
8/i = (8 * -i) / (i * -i) = -8i / (-1) = 8i
So, our first entry in the synthetic division row will be 8i.
Now, we'll proceed further by considering -6i (the coefficient of the second term):
-6/i = (-6 * -i) / (i * -i) = 6i / (-1) = -6i
So, the second entry in the synthetic division row will be -6i.
Continuing this process, we'll have the following entries for the coefficients: 8i, -6i, -57i, -6i, -65i.
Step 4: Perform the synthetic division.
Now, let's perform the synthetic division with the modified coefficients from the previous step:
i | 8 -6 -57 -6 -65
| 8i 2i^2 -55i -49
----------------------------------
| 8i 2i^2 -55i -55
Step 5: Simplify the synthetic division result.
In the remainder row of the synthetic division, we have: 8i + 2i^2 - 55i - 55.
Now, since i^2 = -1, we can rewrite the expression as:
8i + 2(-1) - 55i - 55 = 8i - 2 - 55i - 55 = -57 - 47i.
Step 6: Obtain the quadratic factor.
The remainder obtained in the previous step, -57 - 47i, represents a quadratic factor of the polynomial. To find the remaining factors, we need to factor this quadratic.
To factorize the quadratic factor, we can use the quadratic formula or completing the square method. Let's use the quadratic formula:
The quadratic formula states that for a quadratic equation in the form ax^2 + bx + c = 0, the solutions x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic factor, the coefficients are: a = 2, b = -47, c = -57.
Using the quadratic formula, we have:
x = (-(-47) ± √((-47)^2 - 4(2)(-57))) / (2(2))
= (47 ± √(2209 + 456)) / 4
= (47 ± √(2665)) / 4
Step 7: Simplify the square root.
Now, let's simplify the square root term √(2665). Since this is not a perfect square, we keep it as it is.
Therefore, the quadratic factor can be written as:
x = (47 ± √(2665)) / 4
Step 8: Find the linear factors.
Now that we have determined the quadratic factor, we can write the linear factors by combining it with the complex zeroes, i and -i.
The linear factors are:
x - i = 0 => x = i
x + i = 0 => x = -i
And the quadratic factor:
x = (47 ± √(2665)) / 4
So, the factors of the polynomial 8x^4 - 6x^3 - 57x^2 - 6x - 65, with i as a zero, are (x - i)(x + i)(x - (47 + √(2665))/4)(x - (47 - √(2665))/4).