Find all rational zeros of the polynomial function P(x)=x^4 + 2 x^3 - 13 x^2 + 4 x - 30.
Enter the rational zeros in a comma separated list. If there are none, enter "none".
-3| 1 2 -13 4 -30
-3 15 -6 30
1 -5 2 -10 0
so (x+3)(x^3-5x^2+2x-10
so
5| 1 -5 2 -10
5 0 10
1 0 2
so (x+3)(x-5)(x^2-0x+2)
then sqrt-8
so I got -3,5,i-sqrt2,i+sqrt2
where am I going wrong
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x^4 + 2 x^3 - 13 x^2 + 4 x - 30
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To find the rational zeros of a polynomial function, you can use the Rational Root Theorem. According to the Rational Root Theorem, the possible rational zeros of a polynomial can be obtained by taking the factors of the constant term (in this case -30) and dividing them by the factors of the leading coefficient (in this case 1).
In this case, the possible rational roots can be obtained by taking the factors of -30, which are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30, and dividing them by the factors of 1, which are ±1. So, the possible rational zeros are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.
To determine if any of these values are actually zeros of the polynomial, you can use synthetic division. Based on your explanation, you correctly used synthetic division to test the values -3 and 5 as potential zeros. The result showed that both -3 and 5 are indeed zeros of the polynomial, which means (x + 3) and (x - 5) are factors.
However, after factoring out (x + 3) and (x - 5), you mentioned that you obtained a quadratic term of x^2 + 2x - 10. But this is incorrect. To factorize this quadratic further, you can either apply the quadratic formula or complete the square.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
For the quadratic x^2 + 2x - 10, a = 1, b = 2, c = -10.
Substituting these values into the quadratic formula:
x = (-2 ± √(2^2 - 4(1)(-10)))/(2(1))
Simplifying:
x = (-2 ± √(4 + 40))/2
x = (-2 ± √44)/2
x = (-2 ± 2√11)/2
x = -1 ± √11
So the remaining zeros of the polynomial are -1 + √11 and -1 - √11.
Therefore, the rational zeros of the polynomial function P(x) = x^4 + 2x^3 - 13x^2 + 4x - 30 are -3, 5, -1 + √11, -1 - √11.