Question

F(x)=x² (x−3)(x+1)(x+5) has zeros at x=−5, x=0, and x=3. What is the sign of f on the interval (−[infinity],−5)?

a) Positive
b) Negative
c) Zero
d) None of the above

Answer 1

The sign of f(x)=x²(x-3)(x+1)(x+5) on the interval (-∞,-5) is positive, as the function has an even number of negative factors and one positive factor, which results in a positive product.

Step-by-step explanation:

The function in question is given as f(x)=x²(x-3)(x+1)(x+5). On the interval (-∞,-5), one must consider the signs of each factor of f(x) when x is less than -5. Note that:

• x² is positive since the square of any real number is positive.
• (x-3) is negative because any number less than -5 is also less than 3.
• (x+1) is negative because any number less than -5 is also less than -1.
• (x+5) is negative because we are considering values just to the left of -5.

Since there is an even number of negative signs and the x² term is always positive, the overall sign of f(x) on the interval (-∞,-5) is positive because the negatives cancel out.

Therefore, the sign of f on the interval (-∞,-5) is positive.