To answer this question, we need to consider the relationship between the standard form of a quadratic polynomial and its factored form.
The standard form of a quadratic polynomial is:
[tex]\[ ax^2 + bx + c \][/tex]
The factored form of this polynomial can be written as:
[tex]\[ (x + p)(x + q) \][/tex]
When we expand the factored form [tex]\((x + p)(x + q)\)[/tex], we get:
[tex]\[ x^2 + (p + q)x + pq \][/tex]
By comparing this expanded form to the standard form [tex]\(ax^2 + bx + c\)[/tex], we can make the following observations:
1. The coefficient of [tex]\(x^2\)[/tex] is 1 (assuming [tex]\(a = 1\)[/tex] for simplicity), which matches both forms.
2. The coefficient of [tex]\(x\)[/tex] in the expanded form is [tex]\(p + q\)[/tex], which corresponds to the [tex]\(b\)[/tex] term in the standard form. Therefore, we have:
[tex]\[ b = p + q \][/tex]
3. The constant term in the expanded form is [tex]\(pq\)[/tex], which corresponds to the [tex]\(c\)[/tex] term in the standard form. Therefore, we have:
[tex]\[ c = pq \][/tex]
Now, let's analyze the signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] based on the signs of [tex]\(b\)[/tex] and [tex]\(c\)[/tex]:
1. If [tex]\(c\)[/tex] (i.e., [tex]\(pq\)[/tex]) is positive:
- Both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] must have the same sign because only the product of two numbers with the same sign is positive.
- If [tex]\(b\)[/tex] (i.e., [tex]\(p + q\)[/tex]) is positive, both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] must be positive since their sum is positive.
- If [tex]\(b\)[/tex] is negative, both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] must be negative since their sum is negative.
2. If [tex]\(c\)[/tex] (i.e., [tex]\(pq\)[/tex]) is negative:
- [tex]\(p\)[/tex] and [tex]\(q\)[/tex] must have opposite signs because the product of two numbers with opposite signs is negative.
- If [tex]\(b\)[/tex] (i.e., [tex]\(p + q\)[/tex]) is positive, the positive number between [tex]\(p\)[/tex] and [tex]\(q\)[/tex] has a larger magnitude than the negative number.
- If [tex]\(b\)[/tex] is negative, the negative number between [tex]\(p\)[/tex] and [tex]\(q\)[/tex] has a larger magnitude than the positive number.
In summary, here’s what must be true about the signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
- If [tex]\(c\)[/tex] is positive, [tex]\(p\)[/tex] and [tex]\(q\)[/tex] have the same sign. If [tex]\(b\)[/tex] is positive, [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are both positive. If [tex]\(b\)[/tex] is negative, [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are both negative.
- If [tex]\(c\)[/tex] is negative, [tex]\(p\)[/tex] and [tex]\(q\)[/tex] have opposite signs. The sign of [tex]\(b\)[/tex] will then dictate the relative magnitudes of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]. If [tex]\(b\)[/tex] is positive, the positive number has a larger magnitude. If [tex]\(b\)[/tex] is negative, the negative number has a larger magnitude.
Therefore, the signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] depend on the signs of [tex]\(b\)[/tex] and [tex]\(c\)[/tex].
