Question
All the real zeros of the given polynomial are integers. Find the zeros and write the polynomial in factored form. X^3+2x^2-13x+10
Answers
GPT 3.5
To find the zeros of the polynomial, we can try plugging in integers as potential solutions and see if they make the polynomial equal to zero.
By testing a few integers, we find that x = -2 is a zero of the polynomial.
Since -2 is a zero of the polynomial, we know that (x + 2) is a factor of the polynomial.
To find the remaining factors, we can use polynomial division or synthetic division. Dividing the polynomial by (x + 2), we get:
x^2 - 7x + 5
Now, we need to find the zeros of the new polynomial x^2 - 7x + 5.
By factoring or using the quadratic formula, we find that the zeros are x = 1 and x = 5.
Therefore, the zeros of the original polynomial x^3 + 2x^2 - 13x + 10 are x = -2, x = 1, and x = 5.
To write the polynomial in factored form, we can write it as:
(x + 2)(x - 1)(x - 5)
By testing a few integers, we find that x = -2 is a zero of the polynomial.
Since -2 is a zero of the polynomial, we know that (x + 2) is a factor of the polynomial.
To find the remaining factors, we can use polynomial division or synthetic division. Dividing the polynomial by (x + 2), we get:
x^2 - 7x + 5
Now, we need to find the zeros of the new polynomial x^2 - 7x + 5.
By factoring or using the quadratic formula, we find that the zeros are x = 1 and x = 5.
Therefore, the zeros of the original polynomial x^3 + 2x^2 - 13x + 10 are x = -2, x = 1, and x = 5.
To write the polynomial in factored form, we can write it as:
(x + 2)(x - 1)(x - 5)