1. Consider the leading term of the polynomial function. What is the end behavior of the graph? Describe the end behavior and provide the leading term.
-3x5 + 9x4 + 5x3 + 3
2. Evaluate : sqrt7x(sqrtx-7sqrt7) show your work.
3. Solve the equation. Check for extraneous solutions. Type your answers in the blanks. Show your work.
|4x+3|=9+2x

x = _____ or _____
1. To determine the end behavior of a polynomial function, we look at the leading term of the polynomial. The leading term is the term with the highest degree or exponent. In the given polynomial, the leading term is -3x^5.
Since the leading term has an odd degree and a negative coefficient, the end behavior of the graph will be as follows:
- As x approaches negative infinity (-∞), the graph will decrease without bound.
- As x approaches positive infinity (+∞), the graph will increase without bound.
2. To evaluate the expression sqrt(7x(sqrt(x)-7sqrt(7)), we can simplify it using the properties of square roots:
Step 1: Distribute the square root through the expression:
sqrt(7x*sqrt(x)) - sqrt(7x*7sqrt(7))
Step 2: Simplify each term under the square root:
sqrt(7x)*sqrt(sqrt(x)) - sqrt(7x)*sqrt(7sqrt(7))
Step 3: Simplify further if possible. For the first term, sqrt(7x)*sqrt(sqrt(x)) can be written as sqrt(7x)*sqrt(x^(1/2)), which simplifies to sqrt(7x^3/2). For the second term, sqrt(7x)*sqrt(7sqrt(7)) can be written as sqrt(7x)*sqrt((7sqrt(7))^2), which simplifies to sqrt(7x)*sqrt(49*7) = sqrt(7x)*7sqrt(7) = 7sqrt(7x*7) = 7sqrt(49x).
Step 4: Combine the simplified terms:
sqrt(7x^3/2) - 7sqrt(49x)
This is the final simplified expression.
3. To solve the equation |4x+3|=9+2x, we need to remove the absolute value by considering both positive and negative cases:
First, let's consider the positive case when the expression inside the absolute value is positive:
- Set 4x+3 equal to 9+2x:
4x+3 = 9+2x
Step 1: Simplify the equation:
4x - 2x = 9 - 3
2x = 6
Step 2: Solve for x:
x = 6/2
x = 3
Secondly, let's consider the negative case when the expression inside the absolute value is negative:
- Set 4x+3 equal to the negation of 9+2x:
4x+3 = -(9+2x)
Step 1: Simplify the equation:
4x+3 = -9-2x
Step 2: Combine like terms:
4x + 2x = -9 - 3
6x = -12
Step 3: Solve for x:
x = -12/6
x = -2
So, the two solutions to the equation |4x+3|=9+2x are x = 3 and x = -2.
To check for extraneous solutions, substitute each value back into the original equation and verify if it holds true. In this case, both values satisfy the equation, so there are no extraneous solutions.