To determine whether the polynomial [tex]\(32x^2 + 12x - 57\)[/tex] has real or non-real roots, we need to analyze its discriminant. The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the given polynomial [tex]\(32x^2 + 12x - 57\)[/tex]:
1. Identify the coefficients:
- [tex]\(a = 32\)[/tex]
- [tex]\(b = 12\)[/tex]
- [tex]\(c = -57\)[/tex]
2. Substitute the coefficients into the discriminant formula:
[tex]\[ \Delta = 12^2 - 4 \cdot 32 \cdot (-57) \][/tex]
3. Calculate the discriminant:
- [tex]\(12^2 = 144\)[/tex]
- [tex]\(-4 \cdot 32 \cdot (-57) = -4 \cdot 32 \cdot 57 = -4 \cdot 1824 = -7296\)[/tex]
Thus, the discriminant is:
[tex]\[ \Delta = 144 + 7296 = 7440 \][/tex]
Since the discriminant is a positive number ([tex]\(\Delta = 7440\)[/tex]), this indicates that the quadratic equation has two distinct real roots. Non-real roots occur only when the discriminant is negative.
Therefore, the statement that the polynomial [tex]\(32x^2 + 12x - 57\)[/tex] has two non-real roots is:
B. False
