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To determine the possible signs of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in the factored form [tex]\((x + p)(x + q)\)[/tex] of the polynomial [tex]\( x^2 - 10x + 9 \)[/tex], we first need to identify two key pieces of information: the product of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], and the sum of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
### Step-by-Step Solution
1. Identify the Product of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
In the factored form [tex]\((x + p)(x + q)\)[/tex], when you expand the terms, you get:
[tex]\[ x^2 + (p + q)x + pq \][/tex]
This expanded form corresponds to the standard form [tex]\( ax^2 + bx + c \)[/tex].
Therefore, we compare:
[tex]\[ x^2 + (p + q)x + pq = x^2 - 10x + 9 \][/tex]
From the above equation, we see that the product [tex]\( pq \)[/tex] is equal to the constant term [tex]\( 9 \)[/tex] (the coefficient [tex]\( c \)[/tex]).
2. Identify the Sum of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
The coefficient of the [tex]\( x \)[/tex] term in the expanded form is [tex]\( (p + q) \)[/tex]. Comparing again with the standard form [tex]\( x^2 - 10x + 9 \)[/tex], we observe:
[tex]\[ p + q = -10 \][/tex]
Hence, the sum [tex]\( p + q \)[/tex] is equal to the coefficient of [tex]\( x \)[/tex], which is [tex]\( -10 \)[/tex].
3. Determine the Signs of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- The product [tex]\( pq = 9 \)[/tex] is positive. For the product of two numbers to be positive, either both numbers must be positive, or both must be negative.
- The sum [tex]\( p + q = -10 \)[/tex] is negative. For the sum of two numbers to be negative, both numbers must be negative, since the sum of two positive numbers cannot be negative.
Therefore, for the conditions [tex]\( pq = 9 \)[/tex] (positive product) and [tex]\( p + q = -10 \)[/tex] (negative sum) to hold true simultaneously, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be negative.
### Conclusion
Given the polynomial [tex]\( x^2 - 10x + 9 \)[/tex], the signs of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in the factored form [tex]\((x + p)(x + q)\)[/tex] must both be negative. This is because their product is positive, which means [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same sign, and their sum is negative, which means [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must both be negative.
### Step-by-Step Solution
1. Identify the Product of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
In the factored form [tex]\((x + p)(x + q)\)[/tex], when you expand the terms, you get:
[tex]\[ x^2 + (p + q)x + pq \][/tex]
This expanded form corresponds to the standard form [tex]\( ax^2 + bx + c \)[/tex].
Therefore, we compare:
[tex]\[ x^2 + (p + q)x + pq = x^2 - 10x + 9 \][/tex]
From the above equation, we see that the product [tex]\( pq \)[/tex] is equal to the constant term [tex]\( 9 \)[/tex] (the coefficient [tex]\( c \)[/tex]).
2. Identify the Sum of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
The coefficient of the [tex]\( x \)[/tex] term in the expanded form is [tex]\( (p + q) \)[/tex]. Comparing again with the standard form [tex]\( x^2 - 10x + 9 \)[/tex], we observe:
[tex]\[ p + q = -10 \][/tex]
Hence, the sum [tex]\( p + q \)[/tex] is equal to the coefficient of [tex]\( x \)[/tex], which is [tex]\( -10 \)[/tex].
3. Determine the Signs of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- The product [tex]\( pq = 9 \)[/tex] is positive. For the product of two numbers to be positive, either both numbers must be positive, or both must be negative.
- The sum [tex]\( p + q = -10 \)[/tex] is negative. For the sum of two numbers to be negative, both numbers must be negative, since the sum of two positive numbers cannot be negative.
Therefore, for the conditions [tex]\( pq = 9 \)[/tex] (positive product) and [tex]\( p + q = -10 \)[/tex] (negative sum) to hold true simultaneously, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be negative.
### Conclusion
Given the polynomial [tex]\( x^2 - 10x + 9 \)[/tex], the signs of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in the factored form [tex]\((x + p)(x + q)\)[/tex] must both be negative. This is because their product is positive, which means [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same sign, and their sum is negative, which means [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must both be negative.
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