17.01 Binomial expansions
Expand and simplify:
Find the missing expressions:
\left(t + 2\right) \left(t + 9\right)=t \left(t + 9\right)+ ⬚ \left(⬚\right)
\left(v - 2\right) \left(v - 3\right)=v \left(v - 3\right)-⬚ \left(⬚\right)
\left(t - 2\right) \left(t + 8\right) = t \left(t + 8\right) - ⬚ \left(⬚\right)
State whether the following expressions are equivalent to \left(y - 5\right) \left(y - 8\right):
What expression when multiplied by n + 6 gives n^{2} + 14 n + 48?
Expand:
A student wrote \left(a+b\right)^2=a^2+b^2. Explain why they are incorrect and write the correct answer.
State whether the following expressions are perfect squares or not:
k^{2} + 4
k^{2} - 9 k + 81
k^{2} + 5 k + \dfrac{25}{4}
1^{2} - k + k^{2}
k^{4} + 4 k^{2} + 4
k^{2} + 7 k + \dfrac{49}{4}
100 - 10k + k^{2}
k^{4} + 12k^{2} + 36
Is \left( 3 x + 4\right)^{2} equal to 9 x^{2} + 16? Explain your answer using perfect square expansion.
State whether the following perfect square expansions are correct or not. If not, write its expansion.
\left(3 + 4\right)^{2} = 3^{2} + 4^{2}
\left(y - 3\right)^{2} = y^{2} - 9 y + 9
\left( 2 y + 3\right)^{2} = 4 y^{2} + 12 y + 9
\left(y-7\right)^2=y^2-21y+49
Complete the following:
\left(x - 3\right)^{2} = x^{2} - ⬚ x + ⬚
x^{2} + 64 - ⬚ = \left(x - 8\right)^{2}
\left(x + ⬚\right)^{2} = x^{2} + ⬚ x + 36
\left( - 9 x + 2\right)^{2} = 81 x^{2} - ⬚ x + ⬚
Expand and simplify the following expressions:
\left(a + b\right)^{2}
\left(a - b\right)^{2}
\left(x + 10\right)^{2}
\left(x^{2} + 1\right)^{2}
\left(m - 7\right)^{2}
\left(1 - 9 m\right)^{2}
\left( 10 x + 3\right)^{2}
\left( 5 x + 8 y\right)^{2}
\left( 15 y + \dfrac{1}{5}\right)^{2}
\left(\dfrac{n}{6} - \dfrac{6}{n}\right)^{2}
16 \left( 4 x + 3 y\right)^{2}
\left(x+3\right)^2
\left(m-5\right)^2
\left(s+4t\right)^2
\left(x^2+3\right)^2
\left(4-5m\right)^2
\left(4x+7y\right)^2
\left(6y+\dfrac{1}{2}\right)^2
3 \left( 3 x + 8 y\right)^{2}
9 \left( 3 x + 8 y\right)^{2}
\left(\dfrac{8 x - 9}{9}\right)^{2}
3 x \left( 7 x - 5 y\right)^{2}
\left(t + 5\right)^{2}
\left( \dfrac{x}{4} - \dfrac{4}{x}\right)^{2}
Evaluate the following using perfect square expansion:
202^{2}
596^{2}
402^{2}
499^{2}
Expand the following expressions:
\left(a + b\right) \left(a - b\right)
\left(u + 5\right) \left(u - 5\right)
\left(x - 7\right) \left(x + 7\right)
\left(8 - m\right) \left(m + 8\right)
\left(12 - p\right) \left(12 + p\right)
\left(7 - 6 p\right) \left(7 + 6 p\right)
\left( 3 x - 8\right) \left( 3 x + 8\right)
\left( 8 y + 9\right) \left( 8 y - 9\right)
\left( - 9 y - 1\right) \left( - 9 y + 1\right)
\left( - 7 y - 8\right) \left( - 7 y + 8\right)
\left( 9 x - 2 y\right) \left( 9 x + 2 y\right)
\left( 0.2 x + 3\right) \left( 0.2 x - 3\right)
\left( 2 x - 0.9\right) \left( 2 x + 0.9\right)
\left(v + \dfrac{1}{3}\right) \left(v - \dfrac{1}{3}\right)
\left( 5 x + \dfrac{1}{6}\right) \left( 5 x - \dfrac{1}{6}\right)
\left( 8 x - \dfrac{5}{3}\right) \left( 8 x + \dfrac{5}{3}\right)
\left(x - \dfrac{3}{x}\right) \left(x + \dfrac{3}{x}\right)
Expand and simplify the following expressions:
3 x \left( 5 x - 6 y\right) \left( 5 x + 6 y\right)
3 \left( 9 x - 8 y\right) \left( 9 x + 8 y\right)
a^{2} + \left(9 - a\right) \left(9 + a\right) - 10
\left(\left( 5 t + 9\right) - 6 r\right) \left(\left( 5 t + 9\right) + 6 r\right)
6 \left( 8 x - 9 y\right) \left( 8 x + 9 y\right)
5 x \left( 2 x - 9 y\right) \left( 2 x + 9 y\right)
Evaluate 503 \times 497 by rewriting it as a difference of two squares.
Find the area of a rectangle which has a length of \left(x+4\right)\text{ cm} and a width of \left(x+3\right)\text{ cm}.
A square of side lengths measuring x-1 centimetres has each side enlarged by a factor of 2. Write an expression, in expanded form, for the area of the new square.
If the smallest of three consecutive integers is n, find the product of the three integers in terms of n. Write your answer in expanded form.
Consider the following figure:
Find the area of the large square by multiplying its side lengths. Give your answer in expanded form.
Find the area of the large square by getting the sum of the areas of the smaller rectangles inside it.
Consider the following cube:
Find the surface area of the cube in expanded form.
Find the volume of the cube in expanded form.
Form an expression in expanded form for the shaded area in the given figures:
Consider the rectangular prism shown with its dimensions labelled. Write an expression for the volume, and then expand and fully simplify the result.
