Question

One or more zeros of the polynomial P(x)=x³−3x²−5x−25 are given. Find all remaining zeros.

Answer 1

To find all zeros of the polynomial P(x), identify one zero using factorization or techniques like the Rational Root Theorem, then divide P(x) by (x - a) to get a quadratic equation. Solve this quadratic using the quadratic formula to find the remaining zeros.

Step-by-step explanation:

To find the remaining zeros of the polynomial P(x) = x³ - 3x² - 5x - 25, we need to find at least one zero of the polynomial. However, as none are provided directly, we will first attempt to factor the polynomial or find a factor using the Rational Root Theorem, or synthetic division.

If we do find a zero, let's say 'a', we can then divide the polynomial by (x - a) to obtain a quadratic equation. Once we have the quadratic equation, we can solve for the remaining zeros using the quadratic formula, which is x = [-b ± sqrt(b² - 4ac)] / (2a) where a, b, and c are the coefficients of the equation ax² + bx + c = 0.

If the quadratic does not factor easily, the quadratic formula provides a consistent method for finding the roots of any quadratic equation. After finding all zeros, we can confirm our solutions by substitifying them back into the original polynomial to ensure they satisfy P(x) = 0.

Answer 2

The remaining zeros of the polynomial
`\(P(x)=x^3-3x^2-5x-25\)` are
`\(x=-1\)` and
`\(x=5\)`.

Step-by-step explanation:

To find the remaining zeros of the polynomial
`\(P(x)\)`, we can use the fact that if
`\(r\)` is a zero of
`\(P(x)\)`, then
`\(x-r\)` is a factor of
`\(P(x)\)`. Since one or more zeros are given, let's assume
`\(x=a\)` is a known zero. Now, we can perform synthetic division or polynomial long division to divide
`\(P(x)\)` by
`\(x-a\)` to get the quotient. The result will be a quadratic equation, and we can find its zeros using the quadratic formula or factoring.

Given that
`\(x=-1\)` is a zero, dividing
`\(P(x)\)` by
`\(x+1\)` yields
`\(x^2-4x-25\)`. Factoring this quadratic gives
`\((x+1)(x-5)\)`. Therefore, the remaining zeros are
`\(x=-1\)` and
`\(x=5\)`.

This process utilizes the fundamental theorem of algebra, which states that a polynomial of degree
`\(n\)` has exactly
`\(n\)` complex zeros, counting multiplicities. In this case, since we are given one zero, the division helps in factoring the polynomial and finding the remaining zeros.