Find the remaining factors of the polynomial 2x4−15x3+11x2+78x−40
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To find the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40, we can use polynomial division or synthetic division to divide the polynomial by a known factor.
Let's start by trying to divide by (x - 2) since we can easily guess that x = 2 is a potential root. Using synthetic division, we get:
2 | 2 -15 11 78 -40
-4 2 26 208
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2 -19 13 104 168
The remainder after dividing by (x - 2) is 168. This means that (x - 2) is not a factor of the polynomial.
Now, let's try dividing the polynomial by another potential root, (x + 1):
-1 | 2 -19 13 104 168
-2 21 -34 -70
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2 -21 34 70 98
The remainder after dividing by (x + 1) is 98. This means that (x + 1) is not a factor of the polynomial.
Since we have exhausted the potential rational roots without finding any that work, the remaining factors of the polynomial are the quadratic factors that cannot be further factored using rational roots.
Therefore, the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40 are:
(x - 2), (x + 1), and the quadratic factors that cannot be determined with the given information.
To find the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40, we can use the factor theorem and synthetic division.
Step 1: First, we need to find a possible factor by using the rational roots theorem. According to the theorem, the possible rational roots of the polynomial are the factors of the constant term (in this case, -40) divided by the factors of the leading coefficient (in this case, 2). The factors of -40 are ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40, and the factors of 2 are ±1, ±2.
So, the possible rational roots are:
±1/2, ±1, ±2, ±4, ±5, ±10, ±20, ±40
Step 2: We will use synthetic division to check the possible roots. Let's start with 1/2.
1/2 | 2 -15 11 78 -40
| 1 -7/2 -5/4 41/8
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2 -7/2 3/2 37/4 1/8
The remainder is 1/8, which is not zero, so 1/2 is not a factor of the polynomial.
Step 3: Let's try with -1.
-1 | 2 -15 11 78 -40
| -2 17 -28 10
--------------------------------
2 -17 28 50 -30
The remainder is -30, which is not zero, so -1 is not a factor of the polynomial.
Step 4: Next, we will try with -2.
-2 | 2 -15 11 78 -40
| -4 18 -58 60
--------------------------------
2 -19 29 20 20
The remainder is 20, which is not zero, so -2 is not a factor of the polynomial.
Step 5: We will try the remaining possible factors using synthetic division.
Using -4:
-4 | 2 -15 11 78 -40
| -8 92 -412 1336
---------------------------------
2 -23 103 366 1296
Using 5:
5 | 2 -15 11 78 -40
| 10 -25 18 480
----------------------------------
2 -5 -14 96 440
Using 10:
10 | 2 -15 11 78 -40
| 20 -150 -390 870
---------------------------------
2 5 -139 -312 830
Using -10:
-10 | 2 -15 11 78 -40
| -20 350 -410 1200
--------------------------------
2 10 361 908 1160
Using 20:
20 | 2 -15 11 78 -40
| 40 -50 220 4960
------------------------------
2 25 231 2750 4920
The remainder is not zero for any of the remaining possible factors. This means there are no rational zeros or factors other than the ones found in the synthetic division. Therefore, the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40 are irreducible and cannot be factored further.
To find the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40, we need to first factor out any common factors and then apply polynomial factorization techniques.
Step 1: Look for common factors
In this polynomial, we can see that all the coefficients (2, -15, 11, 78, -40) have a common factor of 2. Let's factor out this common factor:
2x^4 - 15x^3 + 11x^2 + 78x - 40 = 2(x^4 - 7.5x^3 + 5.5x^2 + 39x - 20)
Step 2: Factorize the remaining polynomial
To factorize the remaining polynomial x^4 - 7.5x^3 + 5.5x^2 + 39x - 20, we can use different techniques like grouping, synthetic division, or factoring by grouping. Since this is a fourth-degree polynomial, synthetic division might be more convenient in this case.
Using synthetic division, we can find one factor and then factor the remaining quadratic polynomial:
We start with a guess for a factor (also called a zero) by trying different values for x such as 1, -1, 2, -2, etc., and substitute them into the polynomial until we find a zero.
Let's try with x = 1:
1 | 1 -7.5 5.5 39 -20
| 1 -6.5 -1.5 37.5 19.5
-----------------------------------------
1 -6.5 -1.5 37.5 - 0.5
Since the remainder is not zero, x = 1 is not a zero (factor).
We continue trying different values for x until we find a zero:
Let's try with x = 2:
2 | 1 -7.5 5.5 39 -20
| 2 -11.5 -5 68 236
-----------------------------------------
1 -9.5 -5.5 7 216
Again, the remainder is not zero. We repeat the process.
Let's try with x = -1:
-1 | 1 -7.5 5.5 39 -20
| -1 8.5 -14.5 -24.5 44.5
-----------------------------------------
1 -8.5 -6.5 14.5 24.5
Still not zero. Let's try with x = -2:
-2 | 1 -7.5 5.5 39 -20
| -2 19 -49 -180 320
-----------------------------------------
1 -9.5 9 -141 300
The remainder is still not zero. We can see that none of the values we have tried so far are zeros, which means the polynomial may not have any linear factors.
Step 3: Applying the quadratic formula
Since we were unable to find any linear factors, we can try to factorize the remaining quadratic expression by applying the quadratic formula. The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the solutions (or zeros) are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the quadratic polynomial x^2 - 9.5x + 9, where a = 1, b = -9.5, and c = 9, we can plug these values into the quadratic formula:
x = (-(-9.5) ± √((-9.5)^2 - 4(1)(9))) / (2(1))
x = (9.5 ± √(90.25 - 36)) / 2
x = (9.5 ± √54.25) / 2
x = (9.5 ± 7.36) / 2
Using the positive square root:
x = (9.5 + 7.36) / 2 = 8.43 / 2 = 4.215
Using the negative square root:
x = (9.5 - 7.36) / 2 = 2.14 / 2 = 1.07
So, the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40 are (x - 4.215) and (x - 1.07).