x^4 + x^2 - 20
Help! I have no clue how to factor this! Do i have to square root everything?
(x^2 + 5)(x^2 - 4)
If you set y = x^2 and look at the original as:
y^2 + y - 20
It should factor easily to
(y + 5)(y - 4)
which equals:
(x^2 + 5)(x^2 - 4)
so would the final answer be (x^2 + 5)(x^2 - 4) or (y + 5)(y - 4) ???
The final answer is
(x^2 + 5)(x^2 - 4)
If you multiply that, you will get your original equation.
To factor the given expression x^4 + x^2 - 20, it is not necessary to square root everything. Let me guide you through the process.
1. First, inspect the expression to see if any common factors can be factored out. In this case, there are no common factors.
2. Next, notice that this expression is in the form of a quadratic trinomial, where the variable x is raised to even powers. This suggests that we can use a substitution to simplify the expression.
3. Let's substitute a new variable, such as y, for x^2. Now, our expression can be rewritten as y^2 + y - 20.
4. Factor the quadratic trinomial y^2 + y - 20. To do this, you need to find two numbers that multiply to -20 and add up to 1 (the coefficient of the middle term, which is the coefficient of y).
Pairing factors of -20: (1, -20), (-1, 20), (2, -10), (-2, 10), (4, -5), (-4, 5)
From these pairs, we need to find a pair that sums up to 1. We can see that (-4, 5) is the pair, as -4 + 5 = 1.
Therefore, we can rewrite y^2 + y - 20 as (y - 4)(y + 5).
5. Now that we have factored the quadratic trinomial, replace y with x^2, and rewrite the expression.
(x^2 - 4)(x^2 + 5)
6. At this point, we have factored the original expression as (x^2 - 4)(x^2 + 5). This factorization is not yet complete because we can still factor x^2 - 4 further.
7. Notice that x^2 - 4 is a difference of squares. It can be factored using the formula a^2 - b^2 = (a + b)(a - b). In this case, a = x and b = 2.
Therefore, x^2 - 4 can be factored as (x + 2)(x - 2).
8. Combining the results, we have the complete factorization of the original expression:
(x + 2)(x - 2)(x^2 + 5)
So, the factored form of x^4 + x^2 - 20 is (x + 2)(x - 2)(x^2 + 5).
Remember to always check your work by multiplying the factors together to ensure you obtain the original expression.