list all the possible rational zeros of f using the rational zero theorem. then find all the zeros of the function.
1. f(x)=x^2-6x+5
2. f(x)=x^3+x^2-10x+8
This is just asking you to factor. I'll do the first one for you (I forget how to do problems like the second.) x^2-6x+5..you have to get the two numbers to multiply to your last number (5) and add/subtract up to your middle number (-6), so the numbers you want are -5 and -1, written like this (x-5)(x-1), where x=5 and 1. You can then plug either of those numbers back into the equation and you see that both work.
To find the possible rational zeros of a polynomial function using the Rational Zero Theorem, you need to consider all the factors of the constant term divided by the factors of the leading coefficient. Here's how you can apply the Rational Zero Theorem to find the possible rational zeros of the given functions:
1. f(x) = x^2 - 6x + 5:
- The constant term is 5, and the leading coefficient is 1.
- The possible rational zeros will have the form ±(factors of 5) / (factors of 1).
- The factors of 5 are ±1 and ±5, and the factors of 1 are ±1.
- Therefore, the possible rational zeros are ±1, ±5.
2. f(x) = x^3 + x^2 - 10x + 8:
- The constant term is 8, and the leading coefficient is 1.
- The possible rational zeros will have the form ±(factors of 8) / (factors of 1).
- The factors of 8 are ±1, ±2, ±4, and ±8, and the factors of 1 are ±1.
- Therefore, the possible rational zeros are ±1, ±2, ±4, ±8.
To find all the zeros of the function, you can use various methods like factoring, synthetic division, or graphing. Let's find the zeros of both functions using factorization:
1. f(x) = x^2 - 6x + 5:
- We know the possible rational zeros are ±1, ±5. Let's substitute each value into the function and see if any of them give us a result of zero.
* f(1) = (1)^2 - 6(1) + 5 = 0 ✅
* f(-1) = (-1)^2 - 6(-1) + 5 = 0 ✅
* f(5) = (5)^2 - 6(5) + 5 = 25 - 30 + 5 = 0 ✅
* f(-5) = (-5)^2 - 6(-5) + 5 = 25 + 30 + 5 = 60 ❌
- From the results above, we find that x = 1, x = -1, and x = 5 are the zeros of the function f(x) = x^2 - 6x + 5.
2. f(x) = x^3 + x^2 - 10x + 8:
- We know the possible rational zeros are ±1, ±2, ±4, ±8. Let's substitute each value into the function and check if any of them yield a result of zero.
* f(1) = (1)^3 + (1)^2 - 10(1) + 8 = 1 + 1 - 10 + 8 = 0 ✅
* f(-1) = (-1)^3 + (-1)^2 - 10(-1) + 8 = -1 + 1 + 10 + 8 = 18 ❌
* f(2) = (2)^3 + (2)^2 - 10(2) + 8 = 8 + 4 - 20 + 8 = 0 ✅
* f(-2) = (-2)^3 + (-2)^2 - 10(-2) + 8 = -8 + 4 + 20 + 8 = 24 ❌
* f(4) = (4)^3 + (4)^2 - 10(4) + 8 = 64 + 16 - 40 + 8 = 48 ❌
* f(-4) = (-4)^3 + (-4)^2 - 10(-4) + 8 = -64 + 16 + 40 + 8 = 0 ✅
* f(8) = (8)^3 + (8)^2 - 10(8) + 8 = 512 + 64 - 80 + 8 = 504 ❌
* f(-8) = (-8)^3 + (-8)^2 - 10(-8) + 8 = -512 + 64 + 80 + 8 = 0 ✅
- From the results above, we find that x = 1, x = 2, x = -4, and x = -8 are the zeros of the function f(x) = x^3 + x^2 - 10x + 8.
Therefore, the zeros for each function are as follows:
1. f(x) = x^2 - 6x + 5: x = 1, x = -1, x = 5.
2. f(x) = x^3 + x^2 - 10x + 8: x = 1, x = 2, x = -4, x = -8.