how can i factor this out completely:
x^4 + x^2 – 12
well the answer is 74
(x^2 + 4)(x^2 - 3)
If only integral values are allowed in factoring, you are done.
To factor the expression x^4 + x^2 - 12 completely, you need to find two binomial factors that, when multiplied together, give you x^4 + x^2 - 12.
Step 1: Look for any common factors among the terms. In this case, there are no common factors other than 1.
Step 2: Factor the quadratic expression x^4 + x^2 - 12 into two binomial factors.
To factor a quadratic expression, consider the format (a^2 + 2ab + b^2) - (c^2 + 2cd + d^2).
Match the coefficients of the quadratic expression x^4 + x^2 - 12 with the format above:
a^2 = x^4, b^2 = -12, 2ab = x^2.
Step 3: Use trial and error or the quadratic formula to solve for a, b, c, and d.
From the equation 2ab = x^2, you can see that a = x^2 and b = 2.
Now, match the values of a and b with a^2 + 2ab + b^2 = (a + b)^2:
(a + b)^2 = (x^2 + 2)^2.
To find the second binomial factor, take the square root of -12. The result is -√12, which can be simplified as -2√3:
(c + d)^2 = (c + d)(c + d) = (x^2 - 2√3)(x^2 + 2√3).
Step 4: Write the factored form of the expression x^4 + x^2 - 12:
(x^2 + 2)(x^2 - 2√3)(x^2 + 2√3).
Note: If only integral values are allowed in factoring, then you are done. But if non-integral values are acceptable, you can leave the answer as (x^2 + 2)(x^2 - 2√3)(x^2 + 2√3). Hope that helps!