Oppgaver: Algebra#
Oppgave 1#
Trekk sammen uttrykkene og skriv dem så enkelt som mulig.
a)
\[
3x + 4y - 2x + 5y
\]
\[
x + 9y
\]
\[
(3x - 2x) + (4y + 5y) = x + 9y
\]
b)
\[
5x - 2y + 3x + 4y
\]
\[
8x + 2y
\]
\[
(5x + 3x) + (-2y + 4y) = 8x + 2y
\]
c)
\[
2x + 3y - 4x + 5y
\]
\[
-2x + 8y
\]
\[
(2x - 4x) + (3y + 5y) = -2x + 8y
\]
d)
\[
-x + 2y - 3x + 4y
\]
\[
-4x + 6y
\]
\[
(-x - 3x) + (2y + 4y) = -4x + 6y
\]
Oppgave 2#
Trekk sammen uttrykkene.
a)
\[
2x - (3x + 4y) + 5y
\]
\[
-x + y
\]
\[
2x - (3x + 4y) + 5y = 2x - 3x - 4y + 5y = -x + y
\]
b)
\[
-2x + 4y - (3x - 2y)
\]
\[
-5x + 6y
\]
\[
-2x + 4y - (3x - 2y) = -2x + 4y - 3x + 2y = -5x + 6y
\]
c)
\[
2x - (-3x + 4y) - 3y
\]
\[
5x - 7y
\]
\[
2x - (-3x + 4y) - 3y = 2x + 3x - 4y - 3y = 5x - 7y
\]
d)
\[
x - (2x - 3y) + 4y
\]
\[
-x + 7y
\]
\[
x - (2x - 3y) + 4y = x - 2x + 3y + 4y = -x + 7y
\]
Oppgave 3#
Utvid uttrykkene.
a)
\[
2x(x + 1)
\]
\[
2x^2 + 2x
\]
\[
2x(x + 1) = 2x \cdot x + 2x \cdot 1 = 2x^2 + 2x
\]
b)
\[
3x(x + 2)
\]
\[
3x^2 + 6x
\]
\[
3x(x + 2) = 3x \cdot x + 3x \cdot 2 = 3x^2 + 6x
\]
c)
\[
-2x(x + 4)
\]
\[
-2x^2 - 8x
\]
\[
-2x(x + 4) = -2x \cdot x - 2x \cdot 4 = -2x^2 - 8x
\]
d)
\[
-x(x - 3)
\]
\[
-x^2 + 3x
\]
\[
-x(x - 3) = -x \cdot x + x \cdot 3 = -x^2 + 3x
\]
Oppgave 4#
Faktoriser uttrykkene så mye som mulig.
a)
\[
x^2 + 3x
\]
\[
x(x + 3)
\]
\[
x^2 + 3x = x \cdot x + 3 \cdot x = x(x + 3)
\]
b)
\[
-x^2 + 8x
\]
\[
-x(x - 8)
\]
\[
-x^2 + 8x = -(x^2 - 8x) = -x(x - 8)
\]
c)
\[
2x^2 - 4x
\]
\[
2x(x - 2)
\]
\[
2x^2 - 4x = 2x \cdot x - 2 \cdot 2x = 2x(x - 2)
\]
d)
\[
x^2 - 5x
\]
\[
x(x - 5)
\]
\[
x^2 - 5x = x \cdot x - 5 \cdot x = x(x - 5)
\]
Oppgave 5#
Utvid uttrykkene.
a)
\[
(x + 1)^2
\]
\[
x^2 + 2x + 1
\]
\[
(x + 1)^2 = x^2 + 2\cdot 1 \cdot x + 1^2 = x^2 + 2x + 1
\]
b)
\[
(x - 2)^2
\]
\[
x^2 - 4x + 4
\]
\[
(x - 2)^2 = x^2 - 2\cdot 2 \cdot x + 2^2 = x^2 - 4x + 4
\]
c)
\[
(x + 3)^2
\]
\[
x^2 + 6x + 9
\]
\[
(x + 3)^2 = x^2 + 2\cdot 3 \cdot x + 3^2 = x^2 + 6x + 9
\]
d)
\[
(x - 4)^2
\]
\[
x^2 - 8x + 16
\]
\[
(x - 4)^2 = x^2 - 2\cdot 4 \cdot x + 4^2 = x^2 - 8x + 16
\]
Oppgave 6#
Faktoriser uttrykkene.
a)
\[
x^2 + 10x + 25
\]
\[
(x + 5)^2
\]
\[
x^2 + 10x + 25 = x^2 + 2\cdot x \cdot 5 + 5^2 = (x + 5)^2
\]
b)
\[
x^2 - 12x + 36
\]
\[
(x - 6)^2
\]
\[
x^2 - 12x + 36 = x^2 - 2\cdot 6 \cdot x + 6^2 = (x - 6)^2
\]
c)
\[
4x^2 + 4x + 1
\]
\[
(2x + 1)^2
\]
\[
4x^2 + 4x + 1 = (2x)^2 + 2\cdot 2x \cdot 1 + 1^2 = (2x + 1)^2
\]
d)
\[
4x^2 - 8x + 4
\]
\[
(2x - 2)^2
\]
\[
4x^2 - 8x + 4 = (2x)^2 - 2\cdot 2x \cdot 2 + 2^2 = (2x - 2)^2
\]
Oppgave 7#
Utvid uttrykkene.
a)
\[
(x + 1)(x - 1)
\]
\[
x^2 - 1
\]
\[
(x + 1)(x - 1) = x^2 - 1^2 = x^2 - 1
\]
b)
\[
(x + 2)(x - 2)
\]
\[
x^2 - 4
\]
\[
(x + 2)(x - 2) = x^2 - 2^2 = x^2 - 4
\]
c)
\[
(x + 3)(x - 3)
\]
\[
x^2 - 9
\]
\[
(x + 3)(x - 3) = x^2 - 3^2 = x^2 - 9
\]
d)
\[
(2x + 1)(2x - 1)
\]
\[
4x^2 - 1
\]
\[
(2x + 1)(2x - 1) = (2x)^2 - 1^2 = 4x^2 - 1
\]
Oppgave 8#
Faktoriser uttrykkene.
a)
\[
4x^2 - 4
\]
\[
4(x - 1)(x + 1)
\]
\[
4x^2 - 4 = 4(x^2 - 1) = 4(x - 1)(x + 1)
\]
b)
\[
-x^2 + 4
\]
\[
-(x - 2)(x + 2)
\]
\[
-x^2 + 4 = -(x^2 - 4) = -(x - 2)(x + 2)
\]
c)
\[
-x^2 + 9
\]
\[
-(x - 3)(x + 3)
\]
\[
-x^2 + 9 = -(x^2 - 9) = -(x - 3)(x + 3)
\]
d)
\[
-x^2 + 16
\]
\[
-(x - 4)(x + 4)
\]
\[
-x^2 + 16 = -(x^2 - 16) = -(x - 4)(x + 4)
\]
Oppgave 9#
Faktoriser uttrykkene.
a)
\[
(x - 1)^2 - 9
\]
\[
(x - 4)(x + 2)
\]
\[
(x - 1)^2 - 9 = (x - 1)^2 - 3^2 = (x - 1 - 3)(x - 1 + 3) = (x - 4)(x + 2)
\]
b)
\[
(x + 2)^2 - 16
\]
\[
(x - 2)(x + 6)
\]
\[
(x + 2)^2 - 16 = (x + 2)^2 - 4^2 = (x + 2 - 4)(x + 2 + 4) = (x - 2)(x + 6)
\]
c)
\[
-(x + 2)^2 + 25
\]
\[
-(x - 3)(x + 7)
\]
\[
-(x + 2)^2 + 25 = -\left((x + 2)^2 - 5^2\right) = -(x + 2 - 5)(x + 2 + 5) = -(x - 3)(x + 7)
\]
d)
\[
-(x - 3)^2 + 1
\]
\[
-(x - 4)(x - 2)
\]
\[
-(x - 3)^2 + 1 = -\left((x - 3)^2 - 1^2\right) = -(x - 3 - 1)(x - 3 + 1) = -(x - 4)(x - 2)
\]
Oppgave 10#
Utvid uttrykkene.
a)
\[
(x + 1)(x - 2)
\]
\[
x^2 - x - 2
\]
\[
(x + 1)(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2
\]
b)
\[
(x - 2)(x + 3)
\]
\[
x^2 + x - 6
\]
\[
(x - 2)(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6
\]
c)
\[
-2(x - 5)(x + 4)
\]
\[
-2x^2 + 2x + 40
\]
\[
-2(x - 5)(x + 4) = -2(x^2 + 4x - 5x - 20) = -2(x^2 - x -20) = -2x^2 + 2x + 40
\]
d)
\[
-(x + 1)(x - 4)
\]
\[
-x^2 + 3x + 4
\]
\[
-(x + 1)(x - 4) = -(x^2 - 4x + x - 4) = -(x^2 - 3x - 4) = -x^2 + 3x + 4
\]