Which of the following expressions are factors of 8x^3-125? Select all that apply. B) x+5C) 2x-5D) 2x+5
2.Factor by regrouping.2^3x-4^x+5(2)^x-5
If the factored expression is of the form (A^x+B)(C^DX+E), answer the following questions.
What is A?
What is B?
What is C?
What is D?
What is E?
3.Decide if the grouping of each expression makes it possible to factor by grouping. Select Option A, Option B, both, or neither.
Option A Option B
2x^3-5x^2-8x+20 (2x^3-5x^2)+(-8x+20) (2x^3-8x)+(-5x^2+20)
21xy-12b^2+14xb-18by (21xy-12b^2)+(14xb-18by) (21xy+14xb)+(-12b^2-18by)
m^3-m^2+6m+24 (m^3-m^2)+(6m+24) (m^3+6m)+(-m^-2+24)
4.A space with an area of x^4+2x^3-2x^2-6x-3 units^2 can be enclosed by a frame. What are the possible lengths and widths of the frame if the coefficients of the polynomials are integers? Select all that apply.
A- 1 unit
B- x^2-3 units
D- x^4-2x^2-3 units
E- x^4+2x^3-2x^2-6x-3 units
5.A store sells Hatchimals for $x+2 each. On Friday, the store sold $x^2-4x-12 worth of Hatchimals. Use this information to complete the sentences below with the correct information. The term x^2 OR -4X OR -12 should be rewritten as x+x OR -2x+-2x OR -6x+2x OR -8x+4x OR -6+-6 OR -8+-4 so that the polynomial can be factored by grouping. X+6 OR x-3 OR x-6 Hatchimals were purchased on Friday.
6.Chose the correct information to complete the paragraph proof of the difference of squares identity.
-To prove that the difference of squares, x^2-y^2, can be factored into polynomials with real coefficients, begin with a diagram of a square of side length x OR x^2 OR y OR y^2 with a small square of side length x OR x^2 OR y OR y^2 cut from one corner.
-Cut the diagram into Rectangle #1 with dimension y by x-y OR x by y and Rectangle #2 with dimensions x-y by x-y OR x by x-y.
-Then reconnect the rectangles along the edge with dimension x-y OR y OR x.
-This creates a square with dimension x by x OR a rectangle with dimension x+y by x-y OR a rectangle with dimensions x-y by x-y. Since the area of the original shape has increased OR decreased OR remains constant, x^2-y^2 must equal (x-y)(x+y)
7.Each row gives the area of a polygon. The side lengths of each polygon are polynomials with integer coefficients. Determine what these areas could represent. Select square, rectangle, both or neither.
16x^4-25y^2 square rectangle
X^2+4 square rectangle
x^2-6x+9 square rectangle
8.The volume of a shipping container is x^4-10x^2+9 units^3
If the length is x+1 units, which combination of expressions could represent the height and width? Select all that apply.
9.Determine which expressions are factors of x^4-2x^3-8x^2. Select all that apply.
10.Karen's favorite polynomial is 3x^2-12.
Chris's favorite polynomial is x^3+x^2-6x.
Sydney's favorite polynomial is 2x^3+2x^2-4x.
Which of the following changes would make their GCF become something other than 1?
A) Add x-2 to Sydney's favorite polynomial.
B) Add x+2 to Chris's favorite polynomial
C)Multiply Karens favorite polynomial by x-1.
D) Multiply Sydney's favorite polynomial by x-2.
E) Multiply Chris's favorite polynomial by 3.
11.A tank was filled at a constant rate with water for some amount of time in minutes. In the end, there were x^3+4x^2+x+4 gallons of water in the tank.
Given: gallons/minutes . minutes= gallons
Determine the constant rate in gallon/minute at which the water was poured into the tank and how many minutes it took to fill the tank if both values are polynomials with non-zero integer coefficients and neither polynomial is 1. Select the TWO expressions that represent the constant rate of water and the number in minutes.
12.What is the greatest common factor of the expression 5a^3- 15a^2 and 15a^2- 45a?
Choose the correct terms to factor the expression completely.
3x^3-27x^2+60x=( 3 OR x OR 3x) (x-4 OR x^2-4x OR x+4) (x^2-5x OR x-5 OR x+5)
14.The sum of squares (i.e. x^2+y^2) cannot be factored using polynomials with real coefficients because:
A) The factors would need to be linear binomials with positive coefficients which, when multiplied, would create a quadratic trinomial.
B) there is no greatest common factor other than 1.
C) quadratic binomials cannot be factored
D) binomials cannot be factored.
15.It is possible to factor any differences of squares (i.e. x^2-y^2) using polynomials with real coefficients because:
A) all binomials can be factored
B) The difference of squares always has a greatest common factor other than 1.
C) a difference of terms can always be factored.
D) multiplying linear binomial conjugates (i.e a+b and a-b) will result in the difference of squares.
16.The volume of a rectangular cardboard box is 2x^3+x^2-8x-4 cm^3. its height is x-2 cm and the length and width are polynomials with integer coefficients. Use this information to determine the validity of the information in each row. Select true or false.
- The length and width of the box are x+2 cm and 2x+1 cm. True or False
- The total surface area of the box is 5x^2+2x-4 cm^2. True or false
- it is possible that this box is a cube. True or false
17.Karen spent a total of $3x^4+6x^3-24x^2 on 3x^3-6x^2 jewels. What is the cost of x^2-2x+5 jewels?
The numerator of the simplified version of A(x)/B(x) is (x+3)^3+4(x+3)^2-5(x+3) OR (x+3)(x+8)(x+2) OR (x+3)(x^2+4x-5) OR (x+8)(x+2) and the denominator is (2x+1)(x-5) OR 2x+17 OR 2(x+8)^2-9(x+8)-5 OR 2x+9
X+3 is a factor of A(x) OR B(x) OR A(x) and B(x) OR neither A(x) nor B(x)
x+8 is a factor of A(x) OR B(x) OR A(x) and B(x) OR neither A(x) nor B(x)
2x+1 is a factor of A(x) OR B(x) OR A(x) and B(x) OR neither A(x) nor B(x)
2x+17 is a factor of A(x) OR B(x) OR A(x) and B(x) OR neither A(x) nor B(x)
19.Factor (3x+2)^3+27. If the factored expression is of the form (Ax+B)(Cx^2+Dx+E), answer the following question.
What is A?
What is B?
What is C?
What is D?
What is E?
20.Chose the correct information to complete the paragraph proof of the difference of cubes identify.
To prove that the difference of cubes, x^3-y^3, can be factored into polynomials with real coefficients, begin with a diagram of a cube of side length x OR x^3 OR y OR y^3 with a smaller cube of side length x OR x^3 OR y OR y^3 removed from one corner.
Cut the diagram into three rectangular prisms: one with dimensions x by x by x-y OR x by x by y-x OR x by x by y, one with dimensions y by y by x OR y by y by x+y OR y by y by x-y and one with dimensions x by x-y by x-y OR x by y by x-y OR y by x-y by x-y
The common dimensions of all three rectangle prisms is x OR y OR x-y OR x+y and be factored out so that: x^3-y^3=( x OR y OR x-y OR x+y)(x^2+xy+y^2 OR x^2-xy+y^2 OR 2x^2+xy+2y^2)