The value of the expression √ a^2 + 12 + |b| with a = -2 and b = 14 is 28. The polynomial 6x^4 - 30x^3 - 84x^2 factors completely to 6x^2 (x - 7)(x + 2).
Step-by-step explanation:
To evaluate the expression √ a^2 + 12 + |b| with a = -2 and b = 14, we first need to calculate each part of the expression. The square root of a squared (√ a^2) and the absolute value of b (|b|).
The square root of a number squared is simply the absolute value of the original number. So, √ (-2)^2 equals 2. The absolute value of 14 is 14. Now we can put it all together:
2 + 12 + 14 = 28
For the polynomial 6x^4 - 30x^3 - 84x^2, we factor by finding the greatest common factor (GCF) first, which is 6x^2 here.
6x^2(x^2 - 5x - 14)
The quadratic within the parentheses can be factored further:
x^2 - 5x - 14 = (x - 7)(x + 2)
Thus, the fully factored polynomial is:
6x^2 (x - 7)(x + 2)