Expanding brackets - 1B
1) (x+a)*(x+b)=x^2+a*x+b*x+a*b
a)
x^2+11*x+28
b)
x^2-x-6
c)
x^2-4*x+4
d)
2*x^2+3*x-2*x*y-3*y
e)
4*x^2-x*y+12*x*y-3*y=4*x^2-3*y+11*x*y
f)
6*x^2+2*x*y-12*x*y-4*y^2
=6*x^2-4*y^2-10*x*y
g)
2*x^2-5*x+12
h)
9*x^2+4*y^2+12*x*y
i)
4*x^2+16*x*y+6*x+24*y
j)
2*x^2+3*x*y-5*x+10*x-15*y-25
=2*x^2+3*x*y+5*x-15*y-25
k)
3*x^2-4*x*y-5*x-3*x+4*y+5
=3*x^2-4*x*y-2*x+4*y+5
l)
2*x^2+x*y+5*x-8*x*y-4*y^2-20*y
=2*x^2-4*y^2-7*x*y+5*x-20*y
m)
x^2+2*x*y-x+3*x+6*y-3
=x^2+2*x*y+2*x+6*y-3
n)
2*x^2+2*x*y+3*x+12*x+12*y+18
=2*x^2+2*x*y+15*x+12*y+18
o)
-4*y^2+x*y-3*y+16*y-4*y+12
=4*y^2+x*y+9*y+12
p)
12*x*y-4*y^2+8*y+15*x-5*y+10
=-4*y^2+12*x*y+15*x+3*y+10
q)
5*x*y-2*x^2+3*x-20*y+8*x-12
=-2*x^2+5*x*y+11*x-20*y-12
r)
x*y-5*x-4*y^2+20*y-2*y-10
=-4*y^2+x*y-5*x+18*x-10
2.
a)
(x-4)*(5*x+5)
=5*x^2+5*x-20*x-20
=5*x^2-15*x-20
b)
(2*x+5)*(7*x-2)
=14*x^2-4*x+35*x-10
c)
3*(x^2+6*x+9)
=3*x^2+18*x+27
d)
x*(x^2-y^2)
=x^3-x*y^2
e)
x*(6*x^2+8*x+3*x*y+4*y)
=6*x^3+8*x^2+3*x^2*y+4*x*y
f)
y*(x^2+4*x-5)
=x^2*y+4*x*y-5*y
g)
y*(12*x^2-8*x*y+6*x-4*y)
=12*x^2*y-8*x*y^2-4*y^2+6*x*y
h)
y*(-2*x^2+5*x+14*x-35)
=-2*x^2*y+19*x*y-35*y
i)
x*(10*x^2-4*x+5*x*y-2*y)
=10*x^3+5*x^2*y-2*x*y-4*x^2
j)
x*(x^2+3*x*y+2*x+6*y-8)
=x^3+3*x^2*y+2*x^2+6*x*y-8
k)
y*(2*(x_)^2+x*y-x+10*x+5*y-5)
=2*(x_)^2*y+(x_)*y^2+5*y^2+9*(x_)*y-5*y
l)
y*(3*x-2*y-3+6*x^2-4*x*y-6*x)
+6*x^2*y-4*x*y^2-2*y^2-3*x*y-3*y
m)
x*(2*x^2+3*x+2*x*y+3*y-10*x-15)
=2*x^3+2*x^2*y-7*x^2+3*x*y-15*x
n)
(12*x^2)
3. The diagram shows a rectangle with a square cut out.
The rectangle has length 3*x−y+4 and width x+7.
The square has length x−2.
Find an expanded and simplified expression for the shaded area.
(A_)=(3*x-y+4)*(x+7)-(x-2)^2
=(3*x^2-x*y+25*x-7*y+28)-(x^2-4*x+4)
=2*x^2-x*y+29*x-7*y+24
4. A cuboid has dimensions x+2, 2*x−1 and 2*x+3.
Show that the volume of the cuboid is 4*x^3+12*x^2+5*x−6cm^3.
(x+2cm)*(2*x−1cm)*(2*x+3cm)
=(2*x^2+3*x-2)*(2*x+3)
=4*x^3+*x^2+5*x-6
5. Given that (2*x+5*y)*(3*x−y)*(2*x+y)
=a*x^3+b*x^2*y+c*x*y^2+d*y^3, where a, b, c and d are constants, find the values of a, b, c and d.
(3*x−y)*(2*x+y)*(2*x+5*y)
=(6*x^2-y^2+x*y)*(2*x+5*y)
=12*x^3+32*x^2*y+3*x*y^2-5*y^3
=a*x^3+b*x^2*y+c*x*y^2+d*y^3
∴a,b,c,d=12,32,3,-5
Challenge:
Expand and simplify (x+y)^4.
(x+y)^4
=(x+y)^2×(x+y)^2
=(x^2+2*x*y+y^2)*(x^2+2*x*y+y^2)
=x^4+y^4+4*x^3*y+4*x*y^3+6*x^2*y^2