Explain why x^2 +5x + 8 is prime(not factorable) How do you know?
Give 2 examples of a prime polynomial(can't be both binomials)
Give 2 examples of polynomials that are not prime(they can't be both binomials),factor them,explain what you did to factor them.
How would you explain to another student how to decide if any given polynomial is prime of not? Thanks
a x^2 + b x + c
if sqrt (b^2-4ac) is a whole number then you can factor the quadratic
In this case b^2-4ac = 25 - 32 = -7
sqrt(-7) is an imaginary number so you can not factor this with whole numbers.
To make up some you can factor, do it backwards
for example
(x-1)(x-2) = x^2 -3x + 2
You know you can factor that.
Now one way to make up some you can not factor use big numbers at the end so that 4ac is positive and bigger than b^2
x^2 + 5 x + 100
Damon, Thanks for your help. Appreciate it.
To determine if a polynomial is prime (not factorable), we need to check if it can be factored into two binomials or not.
In the case of x^2 + 5x + 8, we can check if it is factorable by applying the quadratic formula. For a quadratic equation in the form ax^2 + bx + c, the discriminant (b^2 - 4ac) determines whether the equation is factorable or not. If the discriminant is negative, the quadratic equation is not factorable. In the case of x^2 + 5x + 8, the discriminant is 5^2 - 4(1)(8) = 25 - 32 = -7, which is negative. This tells us that x^2 + 5x + 8 is not factorable and hence is a prime polynomial.
Now, let's look at some examples of prime polynomials:
1. x^2 + 7: This is a prime polynomial because it cannot be factored into two binomials. There are no two binomials (in the form (x + a)(x + b)) whose product equals x^2 + 7.
2. 3x^2 + 11x + 5: This polynomial is also prime because it cannot be factored into two binomials. The factors of 3x^2 + 11x + 5 are (3x + 1)(x + 5), but these binomials cannot be further factored.
Now let's consider examples of polynomials that are not prime:
1. x^2 + 6x + 9: This polynomial can be factored by recognizing that it is a perfect square trinomial. We can rewrite it as (x + 3)^2, where (x + 3) is the common factor.
2. 4x^2 - 25: This polynomial can be factored using the difference of squares formula, which states that a^2 - b^2 can be factored as (a + b)(a - b). Applying this formula, we can factor 4x^2 - 25 as (2x + 5)(2x - 5).
To determine if any given polynomial is prime or not, follow these steps:
1. Check if the polynomial can be factored into two binomials. If it can, then it is not prime.
2. Calculate the discriminant by subtracting 4ac from b^2 (where the polynomial is in the form ax^2 + bx + c) or check for specific factorization formulas.
3. If the discriminant is negative or the polynomial cannot be factored into two binomials, then the polynomial is prime (not factorable).
I hope this explanation helps you understand how to determine if a polynomial is prime or not!