Explain why x^2 + 5x + 8 is prime (not factorable). How do you know? Give 2 examples of a prime polynomial (they cannot both be binomials). Give 2 examples of polynomials that are NOT prime (they cannot both be binomials), factor them, and explain what you did to factor them.

How would you explain to another student how to decide if any given polynomial is prime or not?

x^2+5x+25/4 = 25/4-8

(x+5/2)^2=-7/4

Hmmm. a perfect square cannot be a negative number, not in the real domain.

mememme

To determine if the polynomial x^2 + 5x + 8 is prime (not factorable), we can use a factoring technique called the quadratic formula. The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots (or solutions) can be found using the formula x = (-b ± √(b^2 - 4ac))/(2a).

In our case, the polynomial x^2 + 5x + 8 is not equal to 0, so we can set it equal to some other constant, say k, to find its roots. This gives us the equation x^2 + 5x + 8 = k.

If the discriminant (b^2 - 4ac) of the quadratic equation is negative, then the equation has no real roots and is thus prime (not factorable). In our case, the discriminant is 5^2 - 4(1)(8) = 25 - 32 = -7, which is negative. Therefore, x^2 + 5x + 8 is prime.

Two examples of prime polynomials are:

1. x^3 + 2x^2 + 2x + 4: This polynomial is prime because it does not have any common factors that can be factored out.

2. x^5 - x^4 + x^3 - x^2 + x - 1: This polynomial is prime because it cannot be factored into a product of lower degree polynomials.

Two examples of polynomials that are NOT prime are:

1. x^2 + 4x + 3: To factor this polynomial, we need to find two numbers whose sum is 4 and whose product is 3. The numbers that satisfy this condition are 1 and 3. Therefore, we can write the polynomial as (x + 1)(x + 3).

2. x^3 - 8: This polynomial can be factored using the difference of cubes formula, which states that a^3 - b^3 = (a - b)(a^2 + ab + b^2). In our case, the polynomial can be written as (x - 2)(x^2 + 2x + 4).

To explain to another student how to decide if any given polynomial is prime or not, you can follow these steps:

1. Check for common factors: Determine if the polynomial can be factored out by finding any common factors among all the terms. If there are common factors, divide the polynomial by those factors to simplify it.

2. Degree of the polynomial: If the polynomial is of degree 1 (linear), it is considered prime. However, if the polynomial is of degree 2 or higher, further steps are required.

3. Use the quadratic formula or other factoring techniques: If the polynomial is of degree 2, apply the quadratic formula and calculate the discriminant. If the discriminant is negative, the polynomial is prime. Otherwise, you can proceed to factorize it further.

4. Try factoring techniques: Use factoring techniques like factoring by grouping, difference of squares, or the sum/difference of cubes formulas to factorize the polynomial into lower degree polynomials.

By following these steps, you can determine if a polynomial is prime or not.