What are the zeros of the polynomial function 8x⁴+30x³ +35x²+15x+2?
To find the zeros of the polynomial function, we need to find the values of x that make the function equal to zero. We can do this by factoring the polynomial.
To begin, we can check if 2 is a zero of the function by substituting it into the polynomial:
8(2)⁴ + 30(2)³ + 35(2)² + 15(2) + 2 = 128 + 240 + 140 + 30 + 2 = 540
Since the result is not equal to zero, we know that 2 is not a zero of the function.
Next, we can divide the polynomial by (x - 2) using polynomial long division:
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(x - 2) | 8x⁴ + 30x³ + 35x² + 15x + 2
- (8x⁴ - 16x³)
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46x³ + 35x² + 15x + 2
- (46x³ - 92x²)
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127x² + 15x + 2
- (127x² - 254/127x)
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269/127x + 2
- (269/127x - 538/127)
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540/127
The remainder is 540/127, which is not equal to zero.
Therefore, the polynomial function 8x⁴ + 30x³ + 35x² + 15x + 2 does not have any rational zeros.
To find the zeros of the polynomial function 8x⁴+30x³ +35x²+15x+2, we can use various methods, such as factoring, synthetic division, or numerical methods.
Unfortunately, this polynomial does not factor easily. Therefore, we'll resort to finding the zeros numerically using a numerical method like the Newton-Raphson method.
Step 1: Choose an initial guess for the zero. Let's start with x = 1.
Step 2: Use the Newton-Raphson formula to find the next approximation for the zero:
x₁ = x₀ - f(x₀) / f'(x₀)
where x₀ is the initial guess, f(x₀) is the value of the polynomial at x₀, and f'(x₀) is the derivative of the polynomial evaluated at x₀.
Step 3: Repeat Step 2 until the desired accuracy is achieved.
Using this method, we can find the zeros of the polynomial function 8x⁴+30x³ +35x²+15x+2 approximately.
To find the zeros of a polynomial function, we need to solve the equation where the polynomial is equal to zero. In this case, we have the polynomial function 8x⁴ + 30x³ + 35x² + 15x + 2.
To solve for the zeros, we can use various techniques such as factoring, synthetic division, or the use of the quadratic formula.
In this case, the polynomial doesn't easily factor or have any common factors. Therefore, we will use numerical methods to approximate the zeros.
One common numerical method is the Newton-Raphson method. This method uses an initial guess and finds successively better approximations to the zeros.
We can start with an initial guess of x = 1 and apply the Newton-Raphson method to find a more accurate approximation. Let's perform a few iterations:
1. First iteration:
- Substitute x = 1 into the polynomial: 8(1)⁴ + 30(1)³ + 35(1)² + 15(1) + 2 = 90.
- Calculate the derivative of the polynomial: 32x³ + 90x² + 70x + 15.
- Calculate the change in x: Δx = -90 / (32(1)³ + 90(1)² + 70(1) + 15) = -90 / 207.
- Update x: x = 1 + (-90 / 207) = 0.565.
2. Second iteration:
- Substitute x = 0.565 into the polynomial: 8(0.565)⁴ + 30(0.565)³ + 35(0.565)² + 15(0.565) + 2 = 2.215.
- Calculate the derivative of the polynomial at x = 0.565.
- Calculate the change in x: Δx = -2.215 / (derivative at x = 0.565).
- Update x: x = 0.565 + (-2.215 / derivative at x = 0.565).
By repeating these iterations, we can find a more accurate approximation for the zero(s) of the polynomial.