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If (x+k) is a common factor of (x2 + px + q) and (x2 + lx + m), then the value of k is:
- a)l−p
- b)m−q
- c)l−p / m−q
- d)m−q / l−p
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If (x+k) is a common factor of (x2 + px + q) and (x2 + lx + m), then ...
(x + k) is a factor of each one of the given expressions, x = - k will make each zero.
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k2 - pk + q = k2 - Ik + m = 0
k = (m - q) / (l - p)
Most Upvoted Answer
If (x+k) is a common factor of (x2 + px + q) and (x2 + lx + m), then ...
Explanation:
To find the value of k, we need to find the common factor of the two given expressions and compare it with the given options. Let's break down the given expressions and find their common factor.
The first expression is:
x^2 + px + q
The second expression is:
x^2 + lx + m
Step 1: Find the factors of the first expression
To find the factors of x^2 + px + q, we need to find two numbers whose sum is p and product is q. Let's assume the factors are a and b.
So, we have:
a + b = p (Equation 1)
a * b = q (Equation 2)
Step 2: Find the factors of the second expression
To find the factors of x^2 + lx + m, we need to find two numbers whose sum is l and product is m. Let's assume the factors are c and d.
So, we have:
c + d = l (Equation 3)
c * d = m (Equation 4)
Step 3: Find the common factor
Now, we need to find the common factor of (x^2 + px + q) and (x^2 + lx + m). The common factor will be (x - k), where k is the common factor.
So, we have:
(x^2 + px + q) / (x - k) = 0 (Equation 5)
(x^2 + lx + m) / (x - k) = 0 (Equation 6)
Step 4: Simplify the equations
We can simplify equations 5 and 6 by dividing both sides by (x - k).
So, we get:
x + p + (q / (x - k)) = 0 (Equation 7)
x + l + (m / (x - k)) = 0 (Equation 8)
Step 5: Compare the equations
Now, let's compare equations 7 and 8 to find the value of k.
From equation 7, we can see that the common factor (x - k) is present in the denominator, which means (x - k) cannot be equal to zero. Therefore, we can disregard the term (q / (x - k)).
From equation 8, we can see that (x - k) is also present in the denominator, but this time we cannot disregard the term (m / (x - k)).
Therefore, the value of k is given by:
k = m / l - p
Hence, the correct answer is option D: k = m / l - p.
To find the value of k, we need to find the common factor of the two given expressions and compare it with the given options. Let's break down the given expressions and find their common factor.
The first expression is:
x^2 + px + q
The second expression is:
x^2 + lx + m
Step 1: Find the factors of the first expression
To find the factors of x^2 + px + q, we need to find two numbers whose sum is p and product is q. Let's assume the factors are a and b.
So, we have:
a + b = p (Equation 1)
a * b = q (Equation 2)
Step 2: Find the factors of the second expression
To find the factors of x^2 + lx + m, we need to find two numbers whose sum is l and product is m. Let's assume the factors are c and d.
So, we have:
c + d = l (Equation 3)
c * d = m (Equation 4)
Step 3: Find the common factor
Now, we need to find the common factor of (x^2 + px + q) and (x^2 + lx + m). The common factor will be (x - k), where k is the common factor.
So, we have:
(x^2 + px + q) / (x - k) = 0 (Equation 5)
(x^2 + lx + m) / (x - k) = 0 (Equation 6)
Step 4: Simplify the equations
We can simplify equations 5 and 6 by dividing both sides by (x - k).
So, we get:
x + p + (q / (x - k)) = 0 (Equation 7)
x + l + (m / (x - k)) = 0 (Equation 8)
Step 5: Compare the equations
Now, let's compare equations 7 and 8 to find the value of k.
From equation 7, we can see that the common factor (x - k) is present in the denominator, which means (x - k) cannot be equal to zero. Therefore, we can disregard the term (q / (x - k)).
From equation 8, we can see that (x - k) is also present in the denominator, but this time we cannot disregard the term (m / (x - k)).
Therefore, the value of k is given by:
k = m / l - p
Hence, the correct answer is option D: k = m / l - p.
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