Factor each polynomial 4z^2+4z-15
2 answers:
Answer:
(2z+5) (2z-3)
Step-by-step explanation:
First, multiply the first coefficient by the third.
4*-15=-60
Then find factors of -60 that add up to the middle term.
The factors are: -6 and 10-------> -6*10=-60 and -6+10=4 so -6 and 10 are the correct factors.
Break up the middle term with the factors that you got above. The new equation will be: 4-6z+10z-15
Factor by grouping.
First, break the equation into two
(4-6z) (10z-15)
Then, factor out the greatest number you can
2z(2z-3) 5(2z-3)
You are then left with a common binomial of 2z-3. COmbine the common binomial into 1 and combine the 2 numbers you factored out in the step above (2z and 5) The final equation will be this:
(2z+5) (2z-3)
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Answer:
Step-by-step explanation:
1) Use the FOIL method: (a+b)(c+d)=ac+ad+bc+bd.
2) Collect like terms.
3) Simplify.
Therefor, the answer is, 10x^3 - 26x^2 - 12x.
Answer:
(a) Decreasing on (0, 1) and increasing on (1, ∞)
(b) Local minimum at (1, 0)
(c) No inflection point; concave up on (0, ∞)
Step-by-step explanation:
ƒ(x) = x² - x – lnx
(a) Intervals in which ƒ(x) is increasing and decreasing.
Step 1. Find the zeros of the first derivative of the function
ƒ'(x) = 2x – 1 - 1/x = 0
2x² - x -1 = 0
( x - 1) (2x + 1) = 0
x = 1 or x = -½
We reject the negative root, because the argument of lnx cannot be negative.
There is one zero at (1, 0). This is your critical point.
Step 2. Apply the first derivative test.
Test all intervals to the left and to the right of the critical value to determine if the derivative is positive or negative.
(1) x = ½
ƒ'(½) = 2(½) - 1 - 1/(½) = 1 - 1 - 2 = -1
ƒ'(x) < 0 so the function is decreasing on (0, 1).
(2) x = 2
ƒ'(0) = 2(2) -1 – 1/2 = 4 - 1 – ½ = ⁵/₂
ƒ'(x) > 0 so the function is increasing on (1, ∞).
(b) Local extremum
ƒ(x) is decreasing when x < 1 and increasing when x >1.
Thus, (1, 0) is a local minimum, and ƒ(x) = 0 when x = 1.
(c) Inflection point
(1) Set the second derivative equal to zero
ƒ''(x) = 2 + 2/x² = 0
x² + 2 = 0
x² = -2
There is no inflection point.
(2). Concavity
Apply the second derivative test on either side of the extremum.
The function is concave up on (0, ∞).
