For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=x^3+4x^2-5;1$

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = x^3 + 4x^2 - 5\) and the known zero \(x = 1\).
2. Use the fact that if \(x = 1\) is a zero, then \((x - 1)\) is a factor of the polynomial. Perform polynomial division or synthetic division to divide \(f(x)\) by \((x - 1)\).
3. Set up synthetic division with the coefficients of \(f(x)\): 1 (for \(x^3\)), 4 (for \(x^2\)), 0 (for \(x\) term, since it is missing), and -5 (constant term). Divide by the zero 1.
4. After completing the division, write the quotient polynomial, which will be a quadratic. This quadratic represents the remaining factor of \(f(x)\).
5. Find the zeros of the quadratic factor by using factoring, completing the square, or the quadratic formula to determine the other zeros of \(f(x)\).

Key Concepts

Polynomial Zeros and Roots

Zeros or roots of a polynomial are the values of x that make the polynomial equal to zero. Finding all zeros involves identifying all such values, including real and complex roots, which correspond to the x-intercepts of the graph.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial of the form (x - c), especially when a zero c is known. This process simplifies the polynomial to a lower degree, making it easier to find the remaining zeros.

Factoring Quadratic Polynomials

After dividing the polynomial, the resulting quadratic can be factored or solved using the quadratic formula to find the remaining zeros. Factoring involves expressing the quadratic as a product of binomials, revealing its roots.