Find the greatest integral upper bound of the zeros for the following function.
f(x) = 4x^3 - 2x^2 - 12x + 1
Find the real zero(s) to the nearest tenth.
f(x) = 2x^3 + x^2 - 1
[I believe the numbers should be around -1 and 0 and 0 and 1.]
To find the greatest integral upper bound of the zeros for the function f(x) = 4x^3 - 2x^2 - 12x + 1, you can use the Rational Root Theorem.
1. Begin by listing all possible rational roots of the polynomial. These are the numbers that could potentially be solutions to the equation. The possible rational roots are the factors of the constant term (1) divided by the factors of the leading coefficient (4). In this case, the factors of 1 are 1 and -1, and the factors of 4 are 1 and 2. So, the possible rational roots are 1/4, -1/4, 1/2, and -1/2.
2. Using synthetic division, evaluate each of these potential roots one at a time:
For the root 1/4: Perform synthetic division with f(1/4). If the remainder is zero, it means that 1/4 is a root, and the quotient is a lower-degree polynomial.
1/4 | 4 - 2 - 12 + 1
Synthetic division:
1 | 4 -2 -12 1
|____ 1 -1 -4
1 -1 -13 -3
The remainder is -3, so 1/4 is not a root of the polynomial.
For the root -1/4: Perform synthetic division with f(-1/4). If the remainder is zero, it means that -1/4 is a root, and the quotient is a lower-degree polynomial.
-1/4 | 4 - 2 - 12 + 1
Synthetic division:
-1 | 4 -2 -12 1
|____ -4 3 3
4 -6 -9 4
The remainder is 4, so -1/4 is not a root of the polynomial.
For the root 1/2: Perform synthetic division with f(1/2). If the remainder is zero, it means that 1/2 is a root, and the quotient is a lower-degree polynomial.
1/2 | 4 - 2 - 12 + 1
Synthetic division:
1 | 4 -2 -12 1
|____ 2 0 -6
4 -4 -12 -5
The remainder is -5, so 1/2 is not a root of the polynomial.
For the root -1/2: Perform synthetic division with f(-1/2). If the remainder is zero, it means that -1/2 is a root, and the quotient is a lower-degree polynomial.
-1/2 | 4 - 2 - 12 + 1
Synthetic division:
-1 | 4 -2 -12 1
|____ -4 3 4
4 -6 -9 5
The remainder is 5, so -1/2 is not a root of the polynomial.
3. None of the possible rational roots produced a remainder of zero, which means this polynomial doesn't have any integer or rational roots. So, the greatest integral upper bound of the zeros for this function is 0. There are no integer zeros.
Moving on to the second question, to find the real zero(s) to the nearest tenth for the function f(x) = 2x^3 + x^2 - 1, you can use the technique of factoring or graphing.
1. Factoring:
Try to factor the polynomial. Start by looking for common factors among the terms.
2x^3 + x^2 - 1
The expression does not have any common factors, so we cannot factor it directly.
2. Graphing:
You can graph the function to estimate the location of the real zero(s). Plot the points on a graphing calculator or use graphing software.
Graphing the function shows that it crosses the x-axis at approximate x-values of -1 and 0. However, it is difficult to determine the exact values by visual inspection.
3. Use numerical methods:
You can use numerical methods such as the Newton-Raphson method or the bisection method to find a more accurate approximation of the real zero(s). However, since you need the values to the nearest tenth, we can make a guess and then calculate the values using a calculator or software.
Based on the graph or the fact that the function changes sign between -1 and 0, you can estimate that the real zero(s) are approximately -1.0 and 0.0.
Therefore, the real zero(s) to the nearest tenth are approximately -1.0 and 0.0.