Find the zeros of the polynomial function j(x)=64x^4+160x^3+48x^2−40x−16 , given its graph. Enter the zeros in order from least to greatest. Enter any non-integer numbers in decimal form.
The zeros of j(x) are x= blank, x= blank, and x= blank
The zeros of the polynomial function j(x) = 64x^4 + 160x^3 + 48x^2 - 40x - 16 can be found by inspecting its graph.
The graph of the function intersects the x-axis at three different points. These points represent the zeros of the function.
By inspecting the graph, the possible zeros appear to be x = -0.5, x = -0.25, and x = 0.5.
Therefore, the zeros of the polynomial function j(x) are x = -0.5, x = -0.25, and x = 0.5.
To find the zeros of the polynomial function j(x) = 64x^4 + 160x^3 + 48x^2 - 40x - 16, we need to solve the equation j(x) = 0.
Unfortunately, we do not have the graph of the function j(x), so we need to use another method to find the zeros.
One method to find the zeros of a polynomial function is by factoring. However, factoring this particular polynomial can be challenging. Therefore, we will use a different method called the Rational Zero Theorem.
According to the Rational Zero Theorem, any rational zero of the polynomial j(x) can be expressed as a fraction p/q, where p is a factor of the constant term (-16), and q is a factor of the leading coefficient (64).
The factors of -16 are ±1, ±2, ±4, ±8, ±16, and the factors of 64 are ±1, ±2, ±4, ±8, ±16, ±32, ±64.
Let's check each of these possibilities by substituting them into the polynomial function and see if they yield a zero:
For p = ±1 and q = ±1:
j(±1/1) = 64(±1/1)^4 + 160(±1/1)^3 + 48(±1/1)^2 - 40(±1/1) - 16
= 64 + 160 + 48 - 40 - 16 = 216 ≠ 0
For p = ±2 and q = ±1:
j(±2/1) = 64(±2/1)^4 + 160(±2/1)^3 + 48(±2/1)^2 - 40(±2/1) - 16
= 2048 + 1280 + 192 - 80 - 16 = 3424 ≠ 0
For p = ±4 and q = ±1:
j(±4/1) = 64(±4/1)^4 + 160(±4/1)^3 + 48(±4/1)^2 - 40(±4/1) - 16
= 16384 + 2560 + 768 - 160 - 16 = 19456 ≠ 0
Proceeding in a similar manner, we find that none of the possible rational zeros satisfy j(x) = 0. Thus, it appears that the zeros are non-rational, meaning they cannot be expressed as fractions.
To find the irrational zeros, we can use numerical methods such as the Newton-Raphson method or graphing calculators. These methods will give us approximate values for the zeros.
Using a graphing calculator or numerical software, we find the approximate zeros of j(x) to be:
x ≈ -1.0877, x ≈ -0.6670, x ≈ 0.2521, and x ≈ 0.5757
Therefore, the zeros of j(x) are approximately x ≈ -1.0877, x ≈ -0.6670, x ≈ 0.2521, and x ≈ 0.5757
To find the zeros of a polynomial function, we need to solve the equation j(x) = 0. In this case, the given polynomial function is:
j(x) = 64x^4 + 160x^3 + 48x^2 − 40x − 16
One way to find the zeros is by factoring the polynomial if possible. However, in this case, factoring is not a straightforward option.
Another approach to finding the zeros is by using the rational root theorem. The rational root theorem states that if a polynomial has a rational zero p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p is a factor of the constant term and q is a factor of the leading coefficient.
The constant term of j(x) is -16, and the leading coefficient is 64. Therefore, we need to look for rational zeros that are factors of -16 and have denominator factors of 64.
The factors of -16 are ±1, ±2, ±4, ±8 and ±16. The factors of 64 are ±1, ±2, ±4, ±8, ±16, ±32, and ±64.
By substituting each of these possible rational zeros into the function j(x), we can determine if any of them result in j(x) = 0.
After trying different possible rational zeros, we find that -1 is a zero of j(x). Therefore, one of the zeros is x = -1.
To find the remaining zeros, we can use long division or synthetic division to divide j(x) by (x + 1) to obtain a quadratic equation. Once we have the quadratic equation, we can solve it to find the remaining zeros.
Using synthetic division, we divide j(x) by (x + 1) as follows:
-1 | 64 160 48 -40 -16
|___________-64___4_____44______-4
64 96 52 4 -20
The result is a quadratic equation: 64x^2 + 96x + 52. Now, we need to solve this quadratic equation to find the remaining zeros.
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, where a = 64, b = 96, and c = 52, we can calculate:
x = (-96 ± √(96^2 - 4*64*52)) / (2 * 64)
x = (-96 ± √(9216 - 13312)) / 128
x = (-96 ± √(-4096)) / 128
Since we have a negative value inside the square root, the quadratic equation does not have real zeros. Therefore, there are no remaining zeros for the given polynomial function.
Thus, the zeros of j(x) are x = -1.