What is the simplified form of the product Start fraction x squared plus eight x plus fifteen over x minus four end fraction times start fraction x squared minus sixteen over two x plus six end fraction ?
We can factor the first fraction as (x+3)(x+5)/(x-4), and the second fraction as (x-4)(x+4)/(2x+6).
Then, we can simplify by cancelling out common factors:
(x+3)(x+5)/(x-4) * (x-4)(x+4)/(2x+6)
= (x+3)(x+5)/(2x+6) * (x+4)
= (x+3)(x+5)/(2(x+3)) * (x+4)
= (x+5)/2 * (x+4)
= (x+5)(x+4)/2
Therefore, the simplified form of the product is (x+5)(x+4)/2.
Divide: Left Parenthesis negative 7 x plus x-squared plus 15 Right Parenthesis divided by Left Parenthesis negative 3 plus x Right Parenthesis
We can use long division to divide polynomial expressions.
_____________
-3 + x | x^2 + (-7x + 15)
-x^2 + 3x
-----------
-4x + 15
-(-4x + 12)
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3
Therefore, (x^2 + (-7x + 15)) / (−3 + x) = x - 4 + (3 / (-3 + x)).
The final answer can also be expressed as (x-4) - 3/(x-3) or (x-4) + 3/(3-x).
Sarah picks a bushel of apples in 45 min. Andy picks a bushel of apples in 75 min. How long will it take them to pick a bushel together?
A. about 14 min
B. about 28 min
C. about 40 min
D. about 60 min
Let's use the formula:
1/t = 1/t1 + 1/t2
where t is the time it takes for both Sarah and Andy to pick a bushel together, and t1 and t2 are the times it takes each of them individually.
Plugging in the values:
1/t = 1/45 + 1/75
Now we can simplify by finding a common denominator:
1/t = (5/225) + (3/225)
1/t = 8/225
Solving for t:
t = 225/8
t ≈ 28.125
So it would take Sarah and Andy approximately 28 minutes to pick a bushel of apples together.
Therefore, the answer is B. About 28 min.
To find the simplified form of the given expression, let's break it down step-by-step:
Step 1: Factorize the numerator of the first fraction.
Since the quadratic term \(x^2\) is coefficient 1, we can factorize the numerator as follows:
\(x^2 + 8x + 15 = (x + 3)(x + 5)\)
Step 2: Factorize the second fraction.
The numerator \(x^2 - 16\) is a difference of squares, which can be factored as:
\(x^2 - 16 = (x + 4)(x - 4)\)
Step 3: Simplify the denominators of both fractions.
The denominator of the first fraction \(x - 4\) and the denominator of the second fraction \(2x + 6\) do not have common factors that can be canceled out.
Step 4: Combine the two fractions.
\( \frac{{(x + 3)(x + 5)}}{{x - 4}} \times \frac{{(x + 4)(x - 4)}}{{2x + 6}} \)
Step 5: Cancel out the common factors.
Both fractions have a factor of \(x - 4\), which can be canceled out:
\( \frac{{(x + 3)(x + 5)(x + 4)}}{{2x + 6}} \)
Step 6: Simplify further if possible.
The expression cannot be simplified any further, as no common factors exist in the numerator and the denominator.
Therefore, the simplified form of the given expression is \( \frac{{(x + 3)(x + 5)(x + 4)}}{{2x + 6}} \).
To find the simplified form of the given product, let's break it down step by step.
Step 1: Factorize the expressions.
We'll factorize the numerator and denominator of each fraction separately.
Expression 1:
x^2 + 8x + 15 can be factored as (x + 3)(x + 5).
So, the first fraction can be written as (x + 3)(x + 5) / (x - 4).
Expression 2:
x^2 - 16 can be factored as (x - 4)(x + 4).
So, the second fraction can be written as (x - 4)(x + 4) / (2x + 6).
Step 2: Simplify further by canceling out common factors.
In the numerator, we have (x + 3)(x + 5) and (x - 4)(x + 4).
In the denominator, we have (x - 4) and (2x + 6).
By canceling out common factors, we simplify the product as follows:
[(x + 3)(x + 5) / (x - 4)] * [(x - 4)(x + 4) / (2x + 6)]
= [(x + 3)(x + 5)(x + 4)] / [(x - 4)(2x + 6)]
Step 3: Further simplify, if possible.
We can simplify the expression by canceling out common factors between the numerator and denominator. In this case, we can cancel out (x - 4), resulting in:
= (x + 3)(x + 5)(x + 4) / (2x + 6)
Step 4: Simplify the remaining expression, if necessary.
To simplify further, we can divide each term by 2:
= [(x + 3)(x + 5)(x + 4)] / 2(x + 3)
Finally, we can cancel out the common factor of (x + 3):
= (x + 5)(x + 4) / 2
Therefore, the simplified form of the given product is (x + 5)(x + 4) / 2.