What binomial product does each diagram illustrate?
What binomial product does each diagram illustrate?
Simplify.
\displaystyle
4x^2(3x -5y + 8z)
Simplify.
\displaystyle
3m(6m^2-5m + 4) -(4m^3 - 8m^2 + 9)
Expand and simplify.
\displaystyle
(y + 5)(y+ 9)
Expand and simplify.
\displaystyle
(4x -7)(3x + 2)
Expand and simplify.
\displaystyle
(6k + 1)(6k -1)
Expand and simplify.
\displaystyle
(w -8)^2
Expand and simplify.
\displaystyle
(4c + 5d)^2
Expand and simplify.
\displaystyle
2(x -4)(x - 7)-5(8x -9)(8x + 9)
The minimum stopping distance, after a delay of 1 s, for a particular car is
modelled by the formula d = 0.006(s + 1)^2, , where d represents the stopping distance, in metres, and s represents the initial speed, in kilometres per hour.
a) Expand and simplify the formula.
b) Compare the results in both versions of the formula for an initial speed of 60 km/h.
Factor.
\displaystyle
9d^2e^2 +6d^3e
Factor.
\displaystyle
15p^2qr^3-25p^3q^2r + 5pqr
Factor.
\displaystyle
5(x + 6)-2(x + 6)
Factor.
\displaystyle
16x^2 + 8x -6x -3
a) Find an algebraic expression for the surface area of the square—based prism.
b) Expand and simplify your expression from part a).
c) Factor the resulting expression from part b).
Factor.
\displaystyle
x^2 + 11x + 24
Factor.
\displaystyle
y^2 -15y + 56
Factor.
\displaystyle
n^2 - n - 90
Factor.
\displaystyle
x^2 -14x + 49
Factor.
\displaystyle
h^2 -100
Factor.
\displaystyle
d^3 + 16d + 64
Factor fully.
\displaystyle
3k^2 + 12km -36 m^2
Factor fully.
\displaystyle
8y^2 + 19y + 6
Factor fully.
\displaystyle
9w^2-24w + 7
Factor fully.
\displaystyle
25a^2 + 60a + 36
Factor fully.
\displaystyle
121w^2 - 144
Factor fully.
\displaystyle
10x^2 -7xy - 6y^2
Explain how to determine whether or not you can factor 9x^2 - 10x + 18 over the integers.
The area of a rectangle is given as x^2 + 13x - 30.
a) Determine polynomials that represent the length and width of the rectangle.
b) What is the smallest integer value of x for which this area expression makes sense?
Determine all values of k so that each trinomial is a perfect square.
\displaystyle
36x^2 + kx + 121
Determine all values of k so that each trinomial is a perfect square.
\displaystyle
49k^2 -56d + k
Determine all values of k so that each trinomial is a perfect square.
\displaystyle
25x^2 -60xy + ky^2
Determine all values of k so that each trinomial is a perfect square.
\displaystyle
ka^2 +30ab + 9b^2
a) Write an algebraic expression for the area of the shaded region.
b) Write the area expression in factored form.
c) Substitute x = 7 into both forms. Are the results the same? Why?
A parabola has equation \displaystyle
y = 2(x + 6)^2 -2
a) Expand and simplify to write the equation in the form y =ax^2 + bx + c.
b) Factor your equation from part a).
c) Do the three equations represent the same parabola? Justify your response.
The volume of a rectangular prism is given as 9x^3 - 30x^2 + 25x.
a) Determine algebraic expressions for the dimensions.
b) Describe the faces of the prism.
Determine two values of k so that the trinomial can be factored as a difference of squares.
\displaystyle
km^2 -25
Determine two values of k so that the trinomial can be factored as a difference of squares.
\displaystyle
16d^2 -k
Determine two values of k so that the trinomial can be factored as a difference of squares.
\displaystyle
a^2 -kb^2
Factor to evaluate the difference.
\displaystyle
34^2-31^2
Factor to evaluate the difference.
\displaystyle
127^2-126^2
The first three diagrams in a pattern are shown.
a) Write a formula for the total number of small squares in the nth diagram.
b) Write a formula for the number of shaded small squares in the nth diagram.
c) Write a formula for the number of unshaded small squares in the nth diagram.
d) Write your formula from part c) in factored form.
e) Show that both forms of the formula give the same results for the 15th diagram.
a) Find all values of b so that x^2 + bx + 10 can be factored over the integers.
b) Find all values of b so that 4y^2 + by + 5 can be factored over the integers.
c) Write an algebraic expression for the shaded area. Then, write the expression in factored form.