For each polynomial given, use the Rational Root Theorem to list the possible rational zeros. Once you’ve found the possible rational zeros, use the calculator to determine which values are actual zeros of the function. The actual zeros should only be zeros that you mentioned as possible zeros. If you graph the function, you will also see additional zeros, those are NOT be in included below. Function f(x) = x^3 + 2x^2 – 11x – 12 Possible Rational Zeros Actual Rational Zeros Click or tap here to enter text. Click or tap here to enter text. f(x) = x^4 + 4x^3 – 6x^2 – 36x – 27 f(x) = 10x^4 – 3x^3 – 29x^2 + 5x + 12 f(x) = x^4 – 2x^3 – x^2 – 2x – 2 Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text
The Correct Answer and Explanation is:
Here are the correct answers using the Rational Root Theorem and calculator verification:
1. Function: f(x)=x3+2×2−11x−12f(x) = x^3 + 2x^2 – 11x – 12
- Possible Rational Zeros: ±1, ±2, ±3, ±4, ±6, ±12
- Actual Rational Zeros: -3, -1, 4
2. Function: f(x)=x4+4×3−6×2−36x−27f(x) = x^4 + 4x^3 – 6x^2 – 36x – 27
- Possible Rational Zeros: ±1, ±3, ±9, ±27
- Actual Rational Zeros: -3, -1, 3
3. Function: f(x)=10×4−3×3−29×2+5x+12f(x) = 10x^4 – 3x^3 – 29x^2 + 5x + 12
- Possible Rational Zeros: ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2, ±5/2, ±1/5, ±2/5, ±3/5, ±6/5
- Actual Rational Zeros: -1/2, 2, 3
4. Function: f(x)=x4−2×3−x2−2x−2f(x) = x^4 – 2x^3 – x^2 – 2x – 2
- Possible Rational Zeros: ±1, ±2
- Actual Rational Zeros: -1
✍️ Explanation
The Rational Root Theorem is a powerful tool for identifying potential rational roots of a polynomial equation. It states that any rational root of a polynomial equation f(x)=anxn+⋯+a0f(x) = a_nx^n + \dots + a_0 must be of the form pq\frac{p}{q}, where pp is a factor of the constant term a0a_0, and qq is a factor of the leading coefficient ana_n.
To apply this theorem, we first list all factors of the constant term and the leading coefficient. Then, we form all possible combinations of pq\frac{p}{q}, both positive and negative. These are our candidate rational zeros.
Once we have the list of possible rational zeros, we use a calculator or graphing tool to test which of these values actually make the function equal to zero. This can be done by direct substitution or by using graphing technology to identify x-intercepts that match our list.
For example, in the first function f(x)=x3+2×2−11x−12f(x) = x^3 + 2x^2 – 11x – 12, we list ±1, ±2, ±3, ±4, ±6, ±12. Testing these, we find that -3, -1, and 4 are actual zeros. These values satisfy f(x)=0f(x) = 0, confirming they are roots.
This method ensures we only consider rational roots and avoids mistakenly including irrational or complex roots that may appear on a graph. It’s especially useful for factoring polynomials or solving equations algebraically.
