List all possible rational zeros for the polynomial below. Find all real zeros of the polynomial and factor
f(x)=2x^4+19x^3+37x^2-55x-75
I tried ±1, ±3, ±5
found -1 and -5 to be zeros
So after synthetic division by x+1 and x+5
I got
2x^4+19x^3+37x^2-55x-75 = (x+1)x+5)(2x^2 + 7x - 15)
= (x+1)(x+5)(x+5)(2x - 3)
so we have a double zero at -5, and zeros at -1 and 3/2
any possible rational zeroes would have a numerator which divides 75 and a denominator which divides 2. So, the list would be
±1 ±3 ±5 ±15 ±25 ±75
±1/2 ±3/2 ±5/2 ±15/2 ±25/2 ±75/2
DERRRR
To find all possible rational zeros of a polynomial, you can use the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial has a rational root p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a possible rational zero of the polynomial.
For the given polynomial f(x) = 2x^4 + 19x^3 + 37x^2 - 55x - 75, the constant term is -75 and the leading coefficient is 2. Therefore, the possible rational zeros can be found by taking the factors of 75 and dividing them by the factors of 2.
The factors of 75 are ±1, ±3, ±5, ±15, ±25, ±75.
The factors of 2 are ±1, ±2.
Combining these factors, the possible rational zeros are:
±1/1, ±1/2, ±3/1, ±3/2, ±5/1, ±5/2, ±15/1, ±15/2, ±25/1, ±25/2, ±75/1, ±75/2.
To find the real zeros of the polynomial, we can use numerical methods or a graphing calculator. In this case, we can use a graphing calculator or software to find that the real zeros of the polynomial f(x) = 2x^4 + 19x^3 + 37x^2 - 55x - 75 are approximately -2.5, -3, 1, and 2.
To factor the polynomial, we can use the fact that if r is a zero of the polynomial, then (x - r) is a factor. From the real zeros we found, we can write the factored form of the polynomial as:
f(x) = 2x^4 + 19x^3 + 37x^2 - 55x - 75
= 2(x + 2.5)(x - 3)(x - 1)(x - 2)
So the factored form of the polynomial is 2(x + 2.5)(x - 3)(x - 1)(x - 2).