Algebra II Unit 6 Answers PHS
(L6) Solve the system of linear equations. {3x-2y=5 {x+5y=-4
(1,-1)
(L8) Find the solution to the linear system: {3x-5y=-2 {2x+3y=5
(1,1)
(L6) Find the vertex of the parabola. y=3x²-6x+7
(1,4)
(L1) Find the vertex. y=3x²-12x+6
(2,-6)
(L3) Factor the polynomials. 8x³+125
(2x+5)(4x²-10x+25)
(L4) Choose the correct equation represented by the division problem. Left: -2 Above(R1): 2 7 13 14 Above(R2): -4 -6 -14 Underneath: 2 3 7 0
(2x³+7x²+13x+14)÷(x+2)=2x²+3x+7
(Q1) Factor the polynomial. 27x³-64
(3x-4)(9x²+12x+16)
(L4) Choose the correct equation represented by the division problem. Left: 2 Above(R1): 3 5 -14 -18 Above(R2): 6 22 16 Underneath: 3 11 8 -2
(3x³+5x²-14x-18)÷(x-2)=3x²+11x+8-²/ₓ₋₂
(L2) Find the midpoint of AB¯ if A=(3,5) and B=(5,9).
(4,7)
(Q1) Factor the polynomial. x3+8
(x+2)(x²-2x+4)
(L3) Factor the polynomials. x³-27
(x-3)(x²+3x+9)
(L3) Given the center and radius, write the standard form equation of the circle. Center =(3,4), radius =2
(x-3)²+(y-4)²=4
(Q2) Select the correct equation represented by the synthetic division problem. Left: -4 Across(R1): 1 8 14 -8 Across(R2): -4 -16 8 Beneath: 1 4 -2 0
(x³+8x²+14x-8)÷(x+4)=x²+4x-2
(Q2) Select the correct equation represented by the synthetic division problem. Left: 3 Across(R1): 1 1 -14 6 Across(R2): 3 12 -6 Beneath: 1 4 -2 0
(x³+x²-14x+6)÷(x-3)=x²+4x-2
(L7) If x=4-5i and y=-4-7i, what is x+y?
-12i
(L8) Multiply the complex numbers: (3-2i)(-4+5i)
-2+23i
(Q1) Subtract. (5x+2y)-(7x-y)
-2x+3y
(L1) Subtract. (-2x+6)-(x-3)
-3x+9
(L7) If x=3+5i and y=-2+7i, what is y-x?
-5+2i
(L1) Subtract. (4c+1)-(5ab+d)
-5ac+4c-d+1
(L8) Find g∘f(x)=g(f(x)). f(x)=3x+5,g(x)=-2x-3
-6x-13
(L8) Find f∘g(x)=f(g(x)). f(x)=3x+5,g(x)=-2x-3
-6x-4
(L9) Divide the complex numbers. ²⁻³ᶦ/₃₊₂ᵢ
-i
(Q2) Divide using synthetic division. Left: -4 Across: 1 7 2 -40
-x²-3x+10
(L9) Choose the best estimate for the domain of the polynomial function. (-2,0) (-1,2) (0,0)
-∞≤x≤∞
(L9) Choose the best estimate for the domain of the polynomial function. (0,-3) (2,-5) (3,0)
-∞≤x≤∞
(L9) Choose the best estimate for the range of the polynomial function. (-2,0) (-1,2) (0,0)
-∞≤y≤2
(L9) Choose the best estimate for the range of the polynomial function. (0,-3) (2,-5) (3,0)
-∞≤y≤∞
(Q1) What is the fourth row of Pascal's triangle?
1 3 3 1
(L3) The eighth row of Pascal's Triangle is 1 7 21 35 35 21 7 1. What is the ninth row?
1 8 28 56 70 56 28 8 1
(L6) Evaluate the polynomial 8x³+15x²-7x-5 at x=1 using synthetic division.
11
(L8) Find f∘g(x)=f(g(x)). f(x)=3x²,g(x)=2x³
12x⁶
(L6) Evaluate the polynomial 2x⁵-7x⁴-5x³+18x²-1 at x=-1.
13
(L9) Calculate det [1 -4] [2 8]
16
(Q1) Select the term.
2
(Q3) Find the composition of the function g(f(x)). f(x)=4x2+1;g(x)=7x
28x²+7
(L4) Find the quotient. (2x²-3x-5)÷(x+1) Left: -1 Above: 2 -3 -5
2x-5
(Q1) Divide. 6x³/3x
2x²
(L4) Find the quotient. (2x³+x²-11+2)÷(x-2)
2x²+5x-1
(L4) Find the quotient. Left: -3 Above: 2 1 -5 2
2x²-5x+10 Remainder: -28
(L5) Find the real roots of y=x³-3x²+x-3 by inspecting the graph:
3
(L3) Find the x -intercepts of the parabola with equation y=x²-9.
3, -3
(Q3) Find the composition of the function f(g(x)). f(x)=5x-3;g(x)=6x+1
30x+2
(Q2) Divide using synthetic division. Left: 1 Across: 3 8 -2 5
3x²+11x+9 Remainder 14
(L4) Find the quotient. (3x³+17x²+26x+24)÷(x+4)
3x²+5x+6
(Q1) Add. (4a+b)+(2x-y)
4a+b+2x-y
(L8) Find g∘f(x)=g(f(x)). f(x)=3x²,g(x)=2x³
54x⁶
(L4) Find the quotient. Left: 2 Above: 5 3 8 -2
5x²+13x+34 Remainder: 66
(L2) Multiply. (2a)(3x²+2xy-4y³)
6ax²+4axy-8ay³
(L2) Multiply. (3)(2x+5)
6x+15
(L2) Multiply. (3x+1)(2x²+5x-7) (Remember to multiply every term of the binomial by every term of the trinomial, and combine like terms.)
6x³+17x²-16x-7
(Q1) Select the term.
6⁻³
(L3) Solve. |x-3|=4
7, -1
(L1) Add. 2x+5x
7x
(L1) Identify the element of the matrix. X₂,₄= ___ .....[0 2 4 6] X=[0 3 6 9] .....[0 4 8 12]
9
(L2) Multiplying polynomials is done by applying the ___ Property when necessary.
Distributive
(Q3) Choose the best estimate for the domain and range for each graph of the polynomial function. (0,1) (1,0)
Domain: (F.) -∞≤x≤∞ Range: (A.) -∞≤y≤1 [I'm not really sure on either of these]
(Q3) Choose the best estimate for the domain and range for each graph of the polynomial function. (3,0)
Domain: (G.) -∞≤x≤∞ [I think] Range: (H.) 3≤y<∞
(Q2) Use the constant term and leading coefficient of each expression to list all its potential roots. (66/100) 3x⁵-7x⁴-5x³+18x²-5
Factors of Constant: (D.) Factors of Leading Coefficient: (B.) Potential Roots: (A.)
(Q2) Use the constant term and leading coefficient of each expression to list all its potential roots. Select all that apply. x⁵-2x⁴-5x³+9x²+2
Factors of Constant: 1,2,-1,-2 Factors of Leading Coefficient: 1,-1 Potential Roots: ±1, ±2
(L4) Choose the correct graph of y=|x-1|+2.
Intersection Point: (1,2) Opened Upwards
(L3) Match the polynomial with its factored form. (D.) (a-b)(a²+ab+b²) (C.) (a+b)(a²-ab+b²) (B.) (a+b)(a-b) (E.) (a+b)(a+b) (A.) (a-b)(a-b)
Match the polynomial with its factored form. a³-b³ a³+b³ a²-b² a²+2ab+b² a²-2ab+b²
(L7) Apply the horizontal or vertical line test to determine if the inverse of the function will be a function. Is the inverse of the function a function? _____
No
(Q3) Is the inverse of the function a function? ________
No
(L5) Given the equation, y=x²+4, the range, or y -values, will always be greater than or equal to 4, and the graph will never intersect the x -axis. What will be the nature of the roots of y=x²+4?
No real roots
(L3) The triangle of numbers used to find the pattern for any power of binomials is called ___ Triangle.
Pascal's
(Q1) The triangle of numbers used to find the pattern for any power of binomials is called ___ Triangle.
Pascal's
(L5) _____ number roots of a polynomial are the points where the graph of the related polynomial function crosses the x -axis.
Real
(L7) Apply the horizontal or vertical line test to determine if the inverse of the function will be a function. Is the inverse of the function a function? _____ (0,0)
Yes
(Q3) Is the inverse of the function a function? ________ (2,0) (0,5)
Yes
(L4) Find the product. ....[1 2] -2[3 -1] ....[5 0]
[-2 -4] [-6 2] [-10 0]
(L5) Multiply the matrices. ..............[3] [1, 2, 3][1] ..............[-1]
[2] (answer is correct)
(L2) Subtract the matrices. [5 -1]-[2 6] [2 4]-[-3 -4]
[3 -7] [5 8]
(L3) What are the values of a and b? (x+y)⁷=x⁷+7x⁶y+ax⁵y²+35x⁴yb+35x³y⁴+21x²y⁵+7xy⁶+y⁷
a= 21 b= 3
(Q1) What are the values of a and b? x⁴+4xᵇy+ax²y²+4xy³+y⁴
a= 6 b= 3
(L5) If abc=0, then ___.
a=0 OR b=0 OR c=0
(L2) The set of polynomials is ___ closed under multiplication.
always
(L9) If the highest exponent of a polynomial function is odd, then the range of the function is ____ all real numbers.
always
(L9) The domain of a polynomial function is ____ all real numbers.
always
(Q1) The set of polynomials is ___ closed under multiplication.
always
(L2) If the monomial is not zero, the product of a monomial and a polynomial will have ___ the polynomial.
as many terms as
(L2) In most cases, the product of a monomial and a binomial is a ___.
binomial
(L3) The coefficients of the terms on the right side of a ___ equation are the numbers in the triangle.
binomial
(Q1) In most cases, the product of a monomial and a binomial is a ___.
binomial
(Q1) The coefficients of the terms on the right side of a ___ equation are the numbers in Pascal's Triangle.
binomial
(L1) A ___ is the intersection of a plane with one or both nappes of a double cone.
conic section
(Q2) The numerators of any rational roots of a polynomial will be the factors of the ___ term.
constant
(L6) The numerators of any rational roots of a polynomial will be factors of the ___.
constant term
(Q3) The ___ of a polynomial function is always all real numbers.
domain
(Q3) If the highest exponent of a polynomial function is ___, then the range of the function is never all real numbers.
even
(L8) Find f(f(x)) and f(f(f(x))). f(x)=x+3
f(f(x)): (D.) x+6 f(f(f(x))): (A.) x+9
(L8) Find f(f(x)) and f(f(f(x))). f(x)=x
f(f(x)): x f(f(f(x))): x
(L5) What is f(g(x))? f(x)=3x+5 and g(x)=5x-1
f(g(x))=15x+2
(L5) To find the roots of a polynomial, it is often useful to find the ____ of the polynomial.
factors
(Q2) It is often useful to find the ___ of a polynomial in order to find its roots.
factors
(L8) Find f∘g(x)=f(g(x)) and g∘f(x)=g(f(x)). f(x)=3x,g(x)=-2x
f∘g(x)=f(g(x)): -6x g∘f(x)=g(f(x)): -6x
(L8) Find f∘g(x)=f(g(x)) and g∘f(x)=g(f(x)). f(x)=x+8,g(x)=x+4
f∘g(x)=f(g(x)): x+12 g∘f(x)=g(f(x)): x+12
(L7) You can determine if the inverse of a polynomial function is a function by using the ____ line test on the polynomial function.
horizontal
(Q3) You can determine if the inverse of a polynomial function is a function by using the ___ line test on the polynomial function.
horizontal
(L8) Identify the conic section from the equation: 3x²-5y²+2x-6y+17=0
hyperbola
(Q3) The graph of the ___ of a polynomial function is the reflection of the graph of the polynomial function over the line y=x.
inverse
(Q2) The denominators of any rational roots of a polynomial will be the factors of the ___ coefficient.
leading
(L6) The denominators of any rational roots of a polynomial will be factors of the ____.
leading coefficient
(Q3) A(n) ___ function, in the form f(x)=mx+b, is a polynomial function.
linear
(L9) Find the minimum and maximum values for the function with the given domain interval. p(x)=x³, given{x∣x∈ℝ,-9≤x≤7}
minimum value = -729; maximum value = 343
(Q3) If the leading coefficient of a polynomial function is ___, then the right end of the graph always points down.
negative
(L9) If the highest exponent of a polynomial function is even, then the range of the function is ____ all real numbers.
never
(L9) If the leading coefficient of a polynomial function is positive, then the right end of the graph ____ points down.
never
(L2) The set of polynomials is ___ closed under division.
not
(Q1) The set of polynomials is ___ closed under division.
not
(L1) A constant is a real ___.
number
(Q3) If the highest exponent of a polynomial function is ___, then the range of the function is always all real numbers.
odd
(L1) Closure means that whenever you add or subtract two polynomials, you get a ____.
polynomial
(L8) A linear function, in the form f(x)=mx+b is a ___ function.
polynomial
(L8) The composition of a polynomial function and another polynomial function will be a ___ function.
polynomial
(Q1) Closure means that whenever you add or subtract two polynomials, you get a ____.
polynomial
(Q3) The composition of a polynomial function and another polynomial function will be a ___ function.
polynomial
(Q3) If the leading coefficient of a polynomial function is ___, then the right end of the graph always points up.
positive
(L1) A term is a constant, a variable, or a ___ of numbers and variables.
product
(Q3) The ___ of a polynomial function is sometimes all real numbers.
range
(L2) Quotients of polynomials are called ___ expressions.
rational
(L1) A variable is a letter that stands for a(n) ___ number.
real
(Q2) The points where the graph of the polynomial crosses the x -axis are called ___ number roots.
real
(Q2) The value of a polynomial at x=7 is the ___ when the polynomial is divided by x-7.
remainder
(L9) If the leading coefficient of a polynomial function is negative, then the left end of the graph ____ points down.
sometimes
(L9) The range of a polynomial function is ____ all real numbers.
sometimes
(L3) Each number in the triangle is the ___ of the two numbers directly above it in the previous row.
sum
(Q1) Each number in Pascal's Triangle is the ___ of the two numbers directly above it in the previous row.
sum
(Q2) A shortcut method of dividing a polynomial by a linear polynomial by using only the coefficients of the terms of the polynomial is called ___ division.
synthetic
(L4) A shortcut method of dividing a polynomial by a linear polynomial by using only the coefficients of the terms of the polynomial is called ___.
synthetic division
(Q1) If the monomial is not zero, the product of a monomial and a polynomial will have as many ___ as the polynomial.
terms
(L7) You can determine if the inverse of a polynomial function is a function by using the ____ line test on the inverse.
vertical
(Q3) You can determine if the inverse of a polynomial function is a function by using the ___ line test on the inverse.
vertical
(L6) The value of a polynomial at x=1 is the remainder when the polynomial is divided by ____.
x-1
(L4) Find the quotient. (x²-5x+2)÷(x-3) Left: 3 Above: 1 -5 2
x-2 Remainder: -4
(L5) Where does the graph of the polynomial function, f(x)=x²+5x+6, cross the x-axis?
x=-2,-3
(Q2) Choose the correct roots for each polynomial equation. x³+x²-22x-40=(x+2)(x+4)(x-5)
x=-2,-4,5
(L5) Choose the correct roots of the polynomial equation. x³+4x²-11x-30=(x+2)(x-3)(x+5)=0
x=-2,3,-5
(Q2) Choose where the graphs of the polynomial functions would cross the x -axis. f(x)=x²-4x-12
x=-2,6
(Q2) Choose the correct roots for each polynomial equation. x³+2x²-23x-60=(x+3)(x+4)(x-5)
x=-3,-4,5
(Q2) Choose where the graphs of the polynomial functions would cross the x -axis. f(x)=3x²-12x-36
x=-6,2
(L5) Use the quadratic formula x=-b±√b²-4ac/2a to find the roots of the quadratic equation. y=x²+x+1
x=-½±i√³/₂
(Q2) Choose the correct roots for each polynomial equation. x³-7x+6=(x-1)(x+3)(x-2)
x=1,-3,2
(Q2) Choose where the graphs of the polynomial functions would cross the x -axis. f(x)=x²-8x+15
x=5,3
(Q1) Divide. (x³+4x²+8x+8)÷(x+2)
x²+2x+4
(Q2) Divide using synthetic division. Left: 2 Across: 1 2 -5 -6
x²+4x+3
(L8) Find f(f(x)). f(x)=x²
x⁴
(L7) Determine the inverse of the function by interchanging the variables and solving for y in terms of x. y=1/2x - 3/2
y=2x+3
(L7) The graph of the inverse of a polynomial function is the reflection of the graph of the polynomial function over the line ___.
y=x
(L7) Determine the inverse of the function by interchanging the variables and solving for y in terms of x. y=x²-1
y=±√x+1
(L7) Determine the inverse of the function by interchanging the variables and solving for y in terms of x. y=4x²
y=±√ˣ/₂
(Q3) Find the inverse of the given function. y=5x²+1
y=±√ˣ⁻¹/₅
(L7) Determine the inverse of the function by interchanging the variables and solving for y in terms of x. y=2x+3
y=½x-³/₂
(Q3) Find the inverse of the given function. y=3x+2
y=⅓x-⅔
(L2) Convert the equation to slope-intercept form. 4x-5y=15
y=⅘x-3
(L5) Roots of a polynomial are the values that make the polynomial equal to ___.
zero
(Q2) The roots of a polynomial are values that make the polynomial equal ___.
zero
(L6) List all of the potential rational roots of x⁵-7x⁴-5x³+18x²-1.
±1
(L6) List all of the potential rational roots of 2x⁵-7x⁴-5x³+18x²-1.
±1 ±½
(L6) Which of these are NOT potential rational roots of 8x³+15x²-7x-5?
±8
(Q1) Select the term.
½
(L6) An ellipse has the equation ⁽ˣ⁻²⁾²/₃² + ⁽ʸ⁺⁵⁾²/₅² =1 If the ellipse is shifted two spaces to the left and five spaces up, what is the new equation?
ˣ²/₃² + ʸ²/₅² =1
(L2) Divide. Then determine if the final result a polynomial. (6x²+11x+10)÷(3x-2)
• 2x+5 +²⁰/₃x-₂ • no
(L2) Divide. Then determine if the final result a polynomial. (4x³)÷(2x)
• 2x² • yes
(L2) Divide. Then determine if the final result a polynomial. (8x³+5x²+3x-1)÷2x
• 4x²+ ⁵/₂x + ³/₂ -½ • no
(L2) Divide. Then determine if the final result a polynomial. 2x+3⌈8x³+6x²+x+15⌉
• 4x²-3x+5 • yes
(L1) Place a check mark in the box if the expression is a term.
• 5x • -7xy
(L5) Find the roots of the polynomial equation (2x+3)(3x-5)=0.
• ⁵/₃ • -³/₂
(L4) Which is the equation of an ellipse?
⁽ˣ⁻⁵⁾²/₁₆ + ⁽ʸ⁺²⁾²/₉ =1
