Algebra 2 final exam review

4x^2+8x/x^2+5x+6

4x(x+2)/(x+2)(x+3)= 4x/x+3

State if the point (-3, -2) is a solution to the system of equations below x^2+y^2+x-10y-3=0

(-3)2+(-2)2+-3-10-2-3=0 9+4-3+20-3=0 27≠0

125^2/3

25

(5x + 4i)(5x - 4i)

25x^2+16 * remember i^2= -1

Multiply (6n+4)(7n-7)

FOIL 42n^2-14n-28

p+10/ p^2+6p-40

p+10/(p-4)(p+10)= 1/p-4

4x^2=2/x^2=5=2

x= +- 2

Factor 25m^2 +9

(5m+3i)(5m-3i)

24v^6/80 v^7

3/10v

Expand (2𝑥 − 1)^5

32𝑥^5 − 80𝑥^4 + 80𝑥^3 − 40𝑥^2 + 10𝑥 − 1

(6x+7y)^2

36x^2+84xy+49y^2

Find the 3rd term in the expansion of (𝑥 − 𝑦)^3

3xy^2

Simplify x+4/ 8x^2(x-1)^2 + x-5/ 8x^2(x-1)^2

Since these two fractions have a common denominator, just combine the numerators 2x-1/8x^2(x-1)^2

State if the point (-1, 1) is a solution to the system of equations below. x^2+y^2+4x-11y+13=0

To determine if a point is a solution to the system of equations, plug the point into both equations to see if it gives a true statement for BOTH. (-1)2+(1)2+4-1-111+13=0 1+1-4-11+13=0 0 = 0 (-1) - 2(1) + 3 = 0 -1 - 2 + 3 = 0 0 = 0

Factor 16x^2+1

(4x+i)(4x-i)

factor 16y^2+1

(4y+i)(4y-i)

What is the sum of two cubes (a+b)^3

(a+b)(a^2+2ab+b^2) =a^3+3a^2b+3ab^2+b^3

Write a polynomial function of least degree with zeros -1, -3 and -i.

(x + 1), (x + 3), (x + i) and (x - i) are all factors of the polynomial x^4+4x^3+4x^2+4x+3

(-3x^2y^5/x^7y)^3

-27y12/x^15

State the complex conjugate of -4+7i

-4-7i

State the complex conjugate of 9i

-9i

f(x)= x^2+4x-21/x^2-x-20 Find domin vertical asymptote horizontal asymptote holes zeros

Domain: All real numbers except 5 or -4 Vertical asymp: x= 5 and x=-4 horiz. asympt: y=1 holes: none zero: x=-7 x=3

Find the zeros for the polynomial function f(x)=x^3(2x+9)^2 and give the multiplicity for each zero. Indicate whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

First factor: x = 0 So, 0 is a zero with a multiplicity of 3. The graph will cross the x-axis at this point. Second factor: 2x + 9 = 0; x = -9/2 So, -9/2 is a zero with a multiplicity of 2. The graph will touch the x-axis at this point.

Use the graph of the systems of equations below to determine if the point (-7, -4) is a solution to the systems of equations -x^2-2x+4y+51=0 x-2y=1

It looks as if (-7, -4) may be one of the points where the two graphs intersect. Use the intersect feature of your calculator to verify.

Simplify (-13-7n^2-14n^5)- (-4+5n^5-4n^2)

Remember to distribute the negative to all terms in the second parenthesis (-14n^5-5n^5) + (-7n^2+4n^2)+ (-13-4)= -19n^5-3n^2-17

If f(x) has real/rational coefficients and 4, 2i and -1 + i are all zeros, what other numbers must be zeros

Since 2i is a root, -2i must also be a root. Since -1 + i is a root, -1 - i must also be a root.

a sum of two perfect squares is factored just like a difference of two perfect squares, but make sure to include

i with the constant.

How do you write a complex conjugate?

just change the sign in front of the imaginary component.

Evaluate 3x^3+11x^2-5x+1 when x = -4.

plug the given value of x in the polynomial every place there is a variable. So, 3(-4)3+11(-4)2-5-4+1 = 3-64+1116+20+1 = -192+176+20+1 = 5

Expand (𝑦 + 4)^3

𝑦^3 + 12𝑦^2 + 48𝑦 + 64

. A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a constant 2 feet wide. Write a polynomial expression that represents the dimensions of the pool and the surrounding walkway. If a rectangular swimming pool is twice as long as it is wide, let 'w' represent the width and '2w' represent the length. If the walkway is a constant 2 feet wide around the entire pool, that means there is 4 feet added on both the length and the width. So, the new width is w + 4 and the new length is 2w + 4. To write a polynomial that represents the dimensions of the pool, multiply these two together.

(w+4)(2w+4) 2w^2+12w+16

How do you find the zeros of a quadratic function of a graph?

1. see where the graph crosses the x-axis, those are the zeros 2. If it doesn't, then those are considered imaginary roots

How do you do synthetic division

1. take the last number of the factor and divide it into the coefficients 2. Remember, the last number is always the remainder 3. If it is zero then the binomial is a factor of the polynomial, if a number is left over, then it is not a factor

Solve 1/3k^2 = 2k+1/3k^2 - 1/6k^2

1/3k^2 (6k^2) = 2k+1/3k^2 (6k^2) - 1/6k^2 (6k^2) 1(2) = (2K+1) (2) -1 2= 4k+2-1 2= 4k+1 1=4k k=1/4

Simplify (4+2i)(5-3i)

10-12i+-i-6i^2 26-2i

Simpify 5k/7 x 7k-70/k-10

5k/7 [7(k-10)/k-10] = 5k

Multiply (5r+4)(r^2-8r-7)

5r^3-36r^2-67r-28

30x^2/48x

5x/8

Solve for x 5x^2-100=0

5x^2/5=100/5 x^2=20 x= square root 20

Simplify 3+5i/2+i

6-3i+10i-5i^2/4-2i+2i-i^2 11+7i/5

State the conjugate of 6+i

6-i

1024^3/5

64

Simplify 6m+6n/12n^3+ m-n/12n^3

6m+ 6n+m-n/12n^3 7m+5n/12n^3

(3+ square root -4)+(6+square root -16)

9 + 6i

Solve the system of equations below algebraically. -3x^2+2y^2+7x+y-5=0 x-3y=4

In this case, isolate x on the second equation, then use substitution to solve. -3x2+2y2+7x+y-5=0 x=3y+4 -33y+42+2y2+73y+4+y-5=0 -39y2+24y+16+2y2+21y+28+y-5=0 -27y2-72y-48+2y2+21y+28+y-5=0 -25y2-50y-25=0 y2+2y+1=0 y+1y+1=0 y = -1 To solve for x, plug -1 back into the second equation. x = 3(-1) + 4; x = 1 So, (1, -1) is the solution to this system.

Sketch a fourth degree polynomial function with a negative leading coefficient.

In this case, the leading coefficient is negative and degree is even. This tells us the end behavior of the graph falls to the left and falls to the right. Use this information to sketch a graph. Since we aren't given any information about zeros, the graph can cross the axis at any points, or no points at all.

Use the information below to sketch a graph of the polynomial function: Roots: 4 imaginary roots f(x) --> infinity as x -->- infinity f(x) --> infinity as x --> infinity

In this case, we have 4 imaginary roots, which tells us the graph will never cross the x-axis. . The end behavior tells us the graph rises to the left and rises to the right. Use this information to sketch a graph

Sketch a graph with Roots: -4, 1, 3 f(x) --> - infinity as x --> infinity f(x) --> infinity as x --> - infinity

In this case, we have three roots which means the graph crosses the x-axis at the three points -4, 1 and 3. The end behavior tells us the graph rises to the left and falls to the right.

Use the information to sketch a graph of the polynomial function Leading coefficient is negative and highest power is 2 Roots 2,8

In this case, we have two roots which means the graph crosses the x-axis at the two points 2 and 8. The fact that the leading coefficient is negative and degree is even tells us end behavior. In this case, the graph falls to the left and falls to the right. Use this information to sketch a graph

Jason can clean a large tank at an aquarium in about 6 hours. When Jason and Lacy work together, they can clean the tank in about 3.5 hours. About how long would it take Lacy to clean the tank if she worked by herself?

Jason's rate: 1/6 of the tank per hour Lacy's rate: 1 /ℎ of the tank per hour, where h is the number of hours needed to clean the tank by herself Jason's rate x hours worked + Lacy's rate x hours worked = 1 complete job 1 /6 (3.5)+ 1 /ℎ(3.5) = 1 3.5h + 21= 6h 21= 2.5h 8.4=h So, it will take Lacy about 8.4 hours to clean the tank if she worked by herself

Solve 4/x^2 - 2/x=1/x

Multiply all fractions by the LCD (x^2) 4-2x=x 4=3x x=4/3

Determine the end behavior for f(x)= -x^2-6x-10

Since the function has a as a negative leading coefficient and even degree, the end behavior f(x)--> -infinity as x--> - infinity f(x) --> - infinity as x --> + infinity

Simplify 2/x-5 + 2/4(x+1)

Since these fractions do not have a common denominator, we need to find the LCD. In this case, the LCD is 4(x - 5)(x + 1). So, multiply the first fraction by 4(x + 1) on top and bottom, and the second fraction by (x - 5) on top and bottom. 2(4)(x+1)/(x-5)(4)(x+1) + 2(x-5)/4(x+1)(x-5) 8x+8+2x-10/4(x+1)(x-5) = 10x-2/4(x+1)(x-5)

Determine end behavior for f(x) -2x^3 +3x^2 -5x+1

Since this function has a negative leading coefficient and odd degree, the end behavior f(x)--> infinity as x--> - infinity f(x)--> - infinity as x--> infinity

Determine the end behavior for f(x)= -3x+4

Since this function has a negative leading coefficient and odd degree, the end behavior is f(x)--> infinity as x--> - infinity f(x)--> - infinity as x--> infinity

Determine end behavior f(x)= x^5-3x^3+3x+1

Since this function has a positive leading coefficient and odd degree, the end behavior f(x)--> - infinity as x--> - infinity f(x)--> infinity as x --> + infinity

15. Write a polynomial function of least degree with zeros i and 2 + 2i.

Since we're given i and 2 + 2i as roots, we must also remember to include - i and 2 - 2i as roots. (x - i)(x + i)(x - (2 + 2i))(x - (2 - 2i)) x^4+4x^3+9x^2-4x+8

Describe the transformation of f9x)= -2/x-1 +2

The negative represents a reflection over the x -axis. The -1 in the denominator represents a shift 1 units right. The +2 at the end of the equation represents a shift 2 units up. The 2 represents a vertical stretch by a factor of 2.

f(x)= 1/x is the parent function, describe the transformation of f(x)=-3/x-2

The negative represents a reflection over the x-axis. The -2 in the denominator represents a shift 2 units right. The 3 represents a vertical stretch by a factor of 3.

A box without a lid is constructed from a 25 inch x 25 inch piece of cardboard by cutting x inch squares from each corner and folding up the sides. Determine the volume of the box as a function for the variable x, then use a graphing calculator to graph this function.

To determine the volume, multiply length x width x height. The length and width of the box are each (25 - 2x), whereas the height is just x. Multiply these dimensions together to find volume: f(x)=x(25-2x)(25-2x) f(x)= 4x^3-100x^2+625x

Sketch the graph of the function f(x)= (x-2)(x-3)

To determine the zeros, set each factor equal to zero and solve. These are the points where the graph will cross the x-axis. x - 2 = 0; x = 2 x - 3 = 0; x = 3 There are two x's, so the degree is 2 (even) and leading coefficient is positive. This means the graph will rise to the left and rise to the right. Use all of this information to sketch a graph.

Find the zeros for the polynomial function f(x)= (x-2)^2(x+4)^4 and give the multiplicity for each zero. Indicate whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

To find the zeros, take each factor and set it equal to zero. Multiplicity is determined by the exponent on that factor, and whether the graph crosses or touches the x-axis is determined by whether the exponent is even or odd. First factor: x - 2 = 0; x = 2 So, 2 is a zero with a multiplicity of 2. The graph will touch the x-axis at this point. Second factor: x + 4 = 0; x = -4 So, -4 is a zero with a multiplicity of 4. The graph will touch the x-axis at this point.

Solve the system of equations below graphically. x=2-y y=x^2-4

To solve graphically, solve for y on each equation and graph either by hand or using a calculator. If graphing using a calculator, use the intersects feature to find where the two graphs intersect So, the solutions are (2,0) and (-3, 5)

Find the 2nd term in the expansion of (4𝑢 + 1)^3

Use Pascal's triangle 48u^2

(5x^3+26x^2-20x+23) / (x+6)

Use synthetic division -6: 5 26 -20 23 -30 24 -24 _______________________________ 5 -4 4 -1 Since the remainder in this case is -1, the given binomial is NOT a factor of the polynomial.

(v^3+6v^2-8v+64)/ (v+8)

Use synthetic division Since the remainder in this case is 0, the given binomial IS a factor of the polynomial

Simplify 8/x+8 divided by x+8/3x+24

When dividing rational expressions, solve just like multiplying rational expressions. Make sure to multiply by the reciprocal Then, factor what you can, cancel any terms common to the numerator and denominator and simplify 8/x+8 x 3(x+8)/x+8 24/x+8

Simplify 7n^2-42n/n-6 x 1/n-6

When multiplying rational expressions, factor what you can, cancel any terms common to the numerator and denominator, then simplify 7n (n-6)/(n-6) x 1/(n-6) 7n/n-6

Write a polynomial function of least degree with zeros -5 and 3i.

the roots are -5 and 3i, you must include -3i as one of our roots. (x + 5)(x - 3i)(x + 3i) x^3+5x^2+9x+45

Solve the system of equations below algebraically x^2+y^2+16x+y=2 y=2x-2

x2+(2x-2)2+16x+2x-2=2 x2+2x-22x-2+16x+2x-2=2 x2+4x2-8x+4+16x+2x-2=2 5x2+10x=0 5xx+2=0 5x = 0; x = 0 x + 2 = 0; x = -2