Question

Determine the zeros of the polynomial h(x) = x^4 + 3x^3 - 55x^2 - 99x + 630.

Answer 1

The zeros of the polynomial h(x) = x^4 + 3x^3 - 55x^2 - 99x + 630 are x = 1, x = -3, x = -5, and x = 7.

Step-by-step explanation:

The zeros of the polynomial h(x) = x^4 + 3x^3 - 55x^2 - 99x + 630 can be found by factoring or using synthetic division. However, in this case, factoring might be difficult due to the high degree of the polynomial. So, we'll use the Rational Root Theorem and synthetic division to find the zeros of the polynomial.

The Rational Root Theorem states that if a polynomial has a rational root (zero), then it can be expressed as a quotient of two integers: p/q, where p is a factor of the constant term (in this case, 630) and q is a factor of the leading coefficient (in this case, 1).

By testing the factors of 630 (±1, ±2, ±3, ±5, ±6, ±7, ±9, ±10, ±14, ±15, ±18, ±21, ±30, ±35, ±42, ±45, ±63, ±70, ±90, ±105, ±126, ±210, ±315, ±630), we can find the rational zeros (if any) of the polynomial. By synthetic division or long division, we can verify which of the possible rational zeros are actually zeros of the polynomial. The zeros of the polynomial h(x) = x^4 + 3x^3 - 55x^2 - 99x + 630 are x = 1, x = -3, x = -5, and x = 7.

Answer 2

The zeros of the polynomial
`h(x) = x^4 + 3x^3 - 55x^2 - 99x + 630` are x = 3, x = 5, x = -6, and x = -7.

To find the zeros of h(x), we can use various methods such as factoring, synthetic division, or the rational root theorem.

In this case, we will use factoring and synthetic division.

Factor the polynomial

The polynomial
`h(x) = x^4 + 3x^3 - 55x^2 - 99x + 630` can be factored as follows:

`h(x) = (x - 3)(x - 5)(x + 6)(x + 7)`

Set each factor equal to zero

Setting each factor equal to zero gives us the following equations:

x - 3 = 0

x - 5 = 0

x + 6 = 0

x + 7 = 0

Solve for x

Solving each equation gives us the zeros of the polynomial:

x = 3

x = 5

x = -6

x = -7