Find the rational roots for x^4+6x^3+8x^2-5x-10
I know the answer is ±1,±2,±5,±10
but I don’t know the steps I’m supposed to take to find it. can someone help me pls
if they want to know all the possible rational roots, then your answer is correct. There are, of course a maximum of four rational roots to a 4th-degree polynomial.
Review the Rational Root Theorem.
To find the rational roots of a polynomial, you can use the rational root theorem. The rational root theorem states that if a polynomial with integer coefficients has a rational root (p/q), then p must be a factor of the constant term (in this case, -10), and q must be a factor of the coefficient of the leading term (in this case, 1).
Let's apply the rational root theorem to find the rational roots of the given polynomial, x^4 + 6x^3 + 8x^2 - 5x - 10.
Step 1: Determine the factors of the constant term (-10): ±1, ±2, ±5, ±10.
Step 2: Determine the factors of the leading coefficient (1): ±1.
These are the possible rational roots we need to test.
Now, we will test each of these possible roots one by one to see if they are indeed roots of the polynomial.
Step 3: Test ±1 as possible roots:
For x = 1:
Substitute x = 1 into the polynomial: (1)^4 + 6(1)^3 + 8(1)^2 - 5(1) - 10 = 1 + 6 + 8 - 5 - 10 = 0
Since the result is zero, x = 1 is a root of the polynomial.
For x = -1:
Substitute x = -1 into the polynomial: (-1)^4 + 6(-1)^3 + 8(-1)^2 - 5(-1) - 10 = 1 - 6 + 8 + 5 - 10 = -2
Since the result is not zero, x = -1 is not a root of the polynomial.
Step 4: Test ±2 as possible roots:
For x = 2:
Substitute x = 2 into the polynomial: (2)^4 + 6(2)^3 + 8(2)^2 - 5(2) - 10 = 16 + 48 + 32 - 10 - 10 = 76
Since the result is not zero, x = 2 is not a root of the polynomial.
For x = -2:
Substitute x = -2 into the polynomial: (-2)^4 + 6(-2)^3 + 8(-2)^2 - 5(-2) - 10 = 16 - 48 + 32 + 10 - 10 = 0
Since the result is zero, x = -2 is a root of the polynomial.
Step 5: Test ±5 as possible roots:
For x = 5:
Substitute x = 5 into the polynomial: (5)^4 + 6(5)^3 + 8(5)^2 - 5(5) - 10 = 625 + 750 + 200 - 25 - 10 = 1540
Since the result is not zero, x = 5 is not a root of the polynomial.
For x = -5:
Substitute x = -5 into the polynomial: (-5)^4 + 6(-5)^3 + 8(-5)^2 - 5(-5) - 10 = 625 - 750 + 200 + 25 - 10 = 90
Since the result is not zero, x = -5 is not a root of the polynomial.
Step 6: Test ±10 as possible roots:
For x = 10:
Substitute x = 10 into the polynomial: (10)^4 + 6(10)^3 + 8(10)^2 - 5(10) - 10 = 10000 + 6000 + 8000 - 50 - 10 = 22440
Since the result is not zero, x = 10 is not a root of the polynomial.
For x = -10:
Substitute x = -10 into the polynomial: (-10)^4 + 6(-10)^3 + 8(-10)^2 - 5(-10) - 10 = 10000 - 6000 + 8000 + 50 - 10 = 24040
Since the result is not zero, x = -10 is not a root of the polynomial.
From the above steps, we have found that x = 1 and x = -2 are the rational roots of the polynomial x^4 + 6x^3 + 8x^2 - 5x - 10.
To find the rational roots of the polynomial, you can use a method called the Rational Root Theorem. Here are the steps to follow:
Step 1: Write down the coefficients of the polynomial. In this case, the polynomial is:
f(x) = x^4 + 6x^3 + 8x^2 - 5x - 10
Step 2: Determine all the possible rational roots by considering the factors of the constant term (the last term) divided by the factors of the leading coefficient (the coefficient of the highest power).
In this case, the constant term is -10, and the leading coefficient is 1. So the possible rational roots can be found by dividing the factors of -10 (±1, ±2, ±5, ±10) by the factors of 1 (±1).
Step 3: Test the possible rational roots using synthetic division (a simplified form of polynomial long division) or substitution to find out which ones are actually roots.
Start by trying the first possible rational root, let's say x = 1. Substitute x = 1 into the polynomial and perform the calculations:
f(1) = (1)^4 + 6(1)^3 + 8(1)^2 - 5(1) - 10
= 1 + 6 + 8 - 5 - 10
= 0
Since f(1) = 0, it means that x = 1 is a root of the polynomial.
Repeat this process for all the other possible rational roots: -1, 2, -2, 5, -5, 10, -10. Substitute each value into the polynomial and check if the result is equal to zero.
In this case, you will find that x = ±1, ±2, ±5, ±10 are all roots of the polynomial, as you mentioned.
Therefore, the rational roots for the polynomial f(x) = x^4 + 6x^3 + 8x^2 - 5x - 10 are ±1, ±2, ±5, ±10.