[Priya Jethalal]

Advanced Algebra Answer Sheet - Module [7]

Total points earned =

Score
Sample Problem / Insert the problem here using the Equation Editor, such as:
Evaluate 42 –32• 4÷12+5.
Calculations
and Answer / Insert your calculations and answer here.
Faculty Comments / Faculty comments go here.
Problem #1 / Given f(x)=x2+7x-8. Find the vertex, axis, and determine whether there is a maximum or minimum value and calculate that value.
Calculations
and Answer / I.  Find the Vertex:

A= 1, B= 7 and C= -8
X= -b-2a
X= -7/2(1)
X= -7/2
Vertex = [-7/2, f(-7/2) ]
F(-7/2) = (-7/2)^2 + 7(-7/2) – 8
F(-7/2) = -81/4
Vertex = (-7/2 , -81/4)
II.  Find the Axis of Symmetry: x= -7/2
III.  Determine the maximum and minimum value:
X= -b/2a
X = -7/2(1)
X= -7/2
F(-7/2) = (-7/2)^2 + 7(-7/2) – 8
F(-7/2) = 49/4 + 7(-7/2) – 8
F(-7/2) = 49/4 -49/2 – 8
F(-7/2) = (-98 – 32 + 49)/4
F(-7/2) = -81/4
This is a Minimum value because a is greater than 0.
Faculty Comments
Problem #2 / Given fx=2x2-10x+14. Find the vertex, axis, and determine whether there is a maximum or minimum value and calculate that value.
Calculations
and Answer / I.  Find the Vertex:
A= 2, B= -10 and C= 14
X= -b/2a
X= -(-10)/2(2)
X= 10/4
X= 5/2
Vertex = [5/2 , F(5/2)]
F(5/2) = 2(5/2)^2 – 10(5/2) +14
F(5/2) = 3/2
Vertex (5/2, 3/2)
II.  Axis of Symmetry : x= 5/2
III.  Determine the maximum and minimum value:
F(5/2) = 2(5/2)^2 – 10(5/2) +14
F(5/2) = 3/2
This is a minimum value because a is greater than 0
Faculty Comments
Problem #3 / Height of a Projectile. A stone is thrown directly upward from a height of 30ft with an initial velocity of 60ft/sec. The height of the stone t seconds after it has been thrown is given by the function
st=-16t2+60t+30. Determine the time at which the stone reaches its maximum height and find the maximum height.
Calculations
and Answer / st=-16t2+60t+30
A= -16, B= 60 and C= 30
X= -b/2a
X= -60/2(-16)
X= -60/-32
X= 15/8 or 1.875 = 1.9
And
Y= -16(15/8)^2 + 60(15/8) + 30
Y = 345/4 or 86.25 = 86
Therefore, after 1.9 sec. the stone reaches its maximum height of 86ft.
Faculty Comments
Problem #4 / Use synthetic division to evaluate x3+6x2-25x+18x+9.
Calculations
and Answer / x3+6x2-25x+18x+9 = x2-3x+21
= x^2 -3x +2
Faculty Comments
Problem #5 / Use synthetic division to evaluate 2x4+7x3+x-12x+3.
Calculations
and Answer / 2x4+7x3+x-12x+3 = 2x^3 + x^2 – 3x + 10 - 42x+3
Faculty Comments
Problem #6 / Use synthetic division to decide whether k=2, k=-1, or k=0 are zeros of
fx=x4-6x3+x2+24x-20. If not, give the value of f(k).
Calculations
and Answer / fx=x4-6x3+x2+24x-20
-  When k=2
f2=(2)4-623+22+24(2)-20 = 0
yes k=2, is the polynomial fx=x4-6x3+x2+24x-20
- When k= -1
f-1=(-1)4-6-13+-12+24(-1)-20 = -36
No, k= -1 is not the polynomial of
fx=x4-6x3+x2+24x-20
-  When k =0
f0=(0)4-6(0)+02+24(0)-20 = 0
Yes k=0, is the polynomial fx=x4-6x3+x2+24x-20
Faculty Comments
Problem #7 / Factor fx=x4-x3-7x2+x+6 into linear factors given that 1 is a zero of f(x).
Calculations
and Answer / fx=x4-x3-7x2+x+6 = (1)4-13-712+(1)+6 = 0
Use synthetic division method to divide
= x^2 – x – 6
(x+1)= Linear factor of x4-x3-7x2+x+6
Faculty Comments
Problem #8 / Given fx=x3+2x2-13x+10, (a) list all possible rational zeros, (b) find all rational zeros, and (c) factor f(x).
Calculations
and Answer / a)  X= 1, x=-5, x= 2
b)  F(x) =x3+2x2-13x+10
Let p = 10 and q = 1
±p/q
-  Find possible combination of the roots of polynomial function. ±1, ±2, ±5, ±10
-  Substitute into polynomial to find the actual roots.
(1)3+2(1)2-13(1)+10 = 0
x=1 is a root of polynomial.
c)  Since 1 is the known root divide the polynomial (x-1) to find the quotient and the remaining roots.
x3+2x2-13x+10 / x-1
x3+2x2-13x+10 / x-1 =
- Solve the remaining polynomials
f-5= (-5)3+2(-5)2-13(-5)+10 = 0
f(2) = (2)3+2(2)2-13(2)+10 = 0
-  These polynomials can be set into linear factors
(x-1) (x+5) (x-2)
-  These are roots (i.e zeros) of polynomial
x3+2x2-13x+10 , {x= 1,-5,2}
Faculty Comments
Problem #9 / Use Descartes' rule of signs to determine the possible number of positive real zeros and the negative real zeros for fx=x4-9x2-6x+4.
Calculations
and Answer
Faculty Comments
Problem #10 / Find all the complex zeros of fx=5x4-4x3+19x2-16x-4. Give exact values.
Calculations
and Answer
Faculty Comments