# Oppgaver: Algebra :::::::::::::::{exercise} Oppgave 1 --- level: 1 --- Trekk sammen uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 3x + 4y - 2x + 5y $$ ::::{answer} $$ x + 9y $$ :::: ::::{solution} $$ (3x - 2x) + (4y + 5y) = x + 9y $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ 5x - 2y + 3x + 4y $$ ::::{answer} $$ 8x + 2y $$ :::: ::::{solution} $$ (5x + 3x) + (-2y + 4y) = 8x + 2y $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ 2x + 3y - 4x + 5y $$ ::::{answer} $$ -2x + 8y $$ :::: ::::{solution} $$ (2x - 4x) + (3y + 5y) = -2x + 8y $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ -x + 2y - 3x + 4y $$ ::::{answer} $$ -4x + 6y $$ :::: ::::{solution} $$ (-x - 3x) + (2y + 4y) = -4x + 6y $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 2 --- level: 1 --- Trekk sammen uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 2x - (3x + 4y) + 5y $$ ::::{answer} $$ -x + y $$ :::: ::::{solution} $$ 2x - (3x + 4y) + 5y = 2x - 3x - 4y + 5y = -x + y $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ -2x + 4y - (3x - 2y) $$ ::::{answer} $$ -5x + 6y $$ :::: ::::{solution} $$ -2x + 4y - (3x - 2y) = -2x + 4y - 3x + 2y = -5x + 6y $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ 2x - (-3x + 4y) - 3y $$ ::::{answer} $$ 5x - 7y $$ :::: ::::{solution} $$ 2x - (-3x + 4y) - 3y = 2x + 3x - 4y - 3y = 5x - 7y $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ x - (2x - 3y) + 4y $$ ::::{answer} $$ -x + 7y $$ :::: ::::{solution} $$ x - (2x - 3y) + 4y = x - 2x + 3y + 4y = -x + 7y $$ :::: :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} $$ a(b + c) = ab + ac $$ ::: :::::::::::::::{exercise} Oppgave 3 --- level: 1 --- Utvid uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 2x(x + 1) $$ ::::{answer} $$ 2x^2 + 2x $$ :::: ::::{solution} $$ 2x(x + 1) = 2x \cdot x + 2x \cdot 1 = 2x^2 + 2x $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ 3x(x + 2) $$ ::::{answer} $$ 3x^2 + 6x $$ :::: ::::{solution} $$ 3x(x + 2) = 3x \cdot x + 3x \cdot 2 = 3x^2 + 6x $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ -2x(x + 4) $$ ::::{answer} $$ -2x^2 - 8x $$ :::: ::::{solution} $$ -2x(x + 4) = -2x \cdot x - 2x \cdot 4 = -2x^2 - 8x $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ -x(x - 3) $$ ::::{answer} $$ -x^2 + 3x $$ :::: ::::{solution} $$ -x(x - 3) = -x \cdot x + x \cdot 3 = -x^2 + 3x $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 4 --- level: 1 --- Faktoriser uttrykkene så mye som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ x^2 + 3x $$ ::::{answer} $$ x(x + 3) $$ :::: ::::{solution} $$ x^2 + 3x = x \cdot x + 3 \cdot x = x(x + 3) $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ -x^2 + 8x $$ ::::{answer} $$ -x(x - 8) $$ :::: ::::{solution} $$ -x^2 + 8x = -(x^2 - 8x) = -x(x - 8) $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ 2x^2 - 4x $$ ::::{answer} $$ 2x(x - 2) $$ :::: ::::{solution} $$ 2x^2 - 4x = 2x \cdot x - 2 \cdot 2x = 2x(x - 2) $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ x^2 - 5x $$ ::::{solution} $$ x^2 - 5x = x \cdot x - 5 \cdot x = x(x - 5) $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} $$ (a + b)^2 = a^2 + 2ab + b^2 $$ $$ (a - b)^2 = a^2 - 2ab + b^2 $$ ::: :::::::::::::::{exercise} Oppgave 5 --- level: 1 --- Utvid uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (x + 1)^2 $$ ::::{answer} $$ x^2 + 2x + 1 $$ :::: ::::{solution} $$ (x + 1)^2 = x^2 + 2\cdot 1 \cdot x + 1^2 = x^2 + 2x + 1 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (x - 2)^2 $$ ::::{answer} $$ x^2 - 4x + 4 $$ :::: ::::{solution} $$ (x - 2)^2 = x^2 - 2\cdot 2 \cdot x + 2^2 = x^2 - 4x + 4 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ (x + 3)^2 $$ ::::{answer} $$ x^2 + 6x + 9 $$ :::: ::::{solution} $$ (x + 3)^2 = x^2 + 2\cdot 3 \cdot x + 3^2 = x^2 + 6x + 9 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ (x - 4)^2 $$ ::::{answer} $$ x^2 - 8x + 16 $$ :::: ::::{solution} $$ (x - 4)^2 = x^2 - 2\cdot 4 \cdot x + 4^2 = x^2 - 8x + 16 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 6 --- level: 1 --- Faktoriser uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ x^2 + 10x + 25 $$ ::::{answer} $$ (x + 5)^2 $$ :::: ::::{solution} $$ x^2 + 10x + 25 = x^2 + 2\cdot x \cdot 5 + 5^2 = (x + 5)^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ x^2 - 12x + 36 $$ ::::{answer} $$ (x - 6)^2 $$ :::: ::::{solution} $$ x^2 - 12x + 36 = x^2 - 2\cdot 6 \cdot x + 6^2 = (x - 6)^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ 4x^2 + 4x + 1 $$ ::::{answer} $$ (2x + 1)^2 $$ :::: ::::{solution} $$ 4x^2 + 4x + 1 = (2x)^2 + 2\cdot 2x \cdot 1 + 1^2 = (2x + 1)^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ 4x^2 - 8x + 4 $$ ::::{answer} $$ (2x - 2)^2 $$ :::: ::::{solution} $$ 4x^2 - 8x + 4 = (2x)^2 - 2\cdot 2x \cdot 2 + 2^2 = (2x - 2)^2 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} $$ a^2 - b^2 = (a - b)(a + b) $$ ::: :::::::::::::::{exercise} Oppgave 7 --- level: 1 --- Utvid uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (x + 1)(x - 1) $$ ::::{answer} $$ x^2 - 1 $$ :::: ::::{solution} $$ (x + 1)(x - 1) = x^2 - 1^2 = x^2 - 1 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (x + 2)(x - 2) $$ ::::{answer} $$ x^2 - 4 $$ :::: ::::{solution} $$ (x + 2)(x - 2) = x^2 - 2^2 = x^2 - 4 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ (x + 3)(x - 3) $$ ::::{answer} $$ x^2 - 9 $$ :::: ::::{solution} $$ (x + 3)(x - 3) = x^2 - 3^2 = x^2 - 9 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ (2x + 1)(2x - 1) $$ ::::{answer} $$ 4x^2 - 1 $$ :::: ::::{solution} $$ (2x + 1)(2x - 1) = (2x)^2 - 1^2 = 4x^2 - 1 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 8 --- level: 2 --- Faktoriser uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 4x^2 - 4 $$ ::::{answer} $$ 4(x - 1)(x + 1) $$ :::: ::::{solution} $$ 4x^2 - 4 = 4(x^2 - 1) = 4(x - 1)(x + 1) $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ -x^2 + 4 $$ ::::{answer} $$ -(x - 2)(x + 2) $$ :::: ::::{solution} $$ -x^2 + 4 = -(x^2 - 4) = -(x - 2)(x + 2) $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ -x^2 + 9 $$ ::::{answer} $$ -(x - 3)(x + 3) $$ :::: ::::{solution} $$ -x^2 + 9 = -(x^2 - 9) = -(x - 3)(x + 3) $$ :::: :::: ::::::::::::: :::::::::::::{tab-item} d $$ -x^2 + 16 $$ ::::{answer} $$ -(x - 4)(x + 4) $$ :::: ::::{solution} $$ -x^2 + 16 = -(x^2 - 16) = -(x - 4)(x + 4) $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Tips: Oppgave 9 I oppgave 9a kan du bruke konjugatsetningen $$ a^2 - b^2 = (a - b)(a + b) $$ ved å sette $$ a^2 = (x - 1)^2 \qog b = 3^2 $$ ::: :::::::::::::::{exercise} Oppgave 9 --- level: 3 --- Faktoriser uttrykkene med konjugatsetningen. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (x - 1)^2 - 9 $$ ::::{answer} $$ (x - 4)(x + 2) $$ :::: ::::{solution} $$ (x - 1)^2 - 9 = (x - 1)^2 - 3^2 = (x - 1 - 3)(x - 1 + 3) = (x - 4)(x + 2) $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (x + 2)^2 - 16 $$ ::::{answer} $$ (x - 2)(x + 6) $$ :::: ::::{solution} $$ (x + 2)^2 - 16 = (x + 2)^2 - 4^2 = (x + 2 - 4)(x + 2 + 4) = (x - 2)(x + 6) $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ -(x + 2)^2 + 25 $$ ::::{answer} $$ -(x - 3)(x + 7) $$ :::: ::::{solution} $$ -(x + 2)^2 + 25 = -\left((x + 2)^2 - 5^2\right) = -(x + 2 - 5)(x + 2 + 5) = -(x - 3)(x + 7) $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ -(x - 3)^2 + 1 $$ ::::{answer} $$ -(x - 4)(x - 2) $$ :::: ::::{solution} $$ -(x - 3)^2 + 1 = -\left((x - 3)^2 - 1^2\right) = -(x - 3 - 1)(x - 3 + 1) = -(x - 4)(x - 2) $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} $$ (a + b)(c + d) = ac + ad + bc + bd $$ ::: :::::::::::::::{exercise} Oppgave 10 --- level: 3 --- Utvid uttrykkene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (x + 1)(x - 2) $$ ::::{answer} $$ x^2 - x - 2 $$ :::: ::::{solution} $$ (x + 1)(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (x - 2)(x + 3) $$ ::::{answer} $$ x^2 + x - 6 $$ :::: ::::{solution} $$ (x - 2)(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ -2(x - 5)(x + 4) $$ ::::{answer} $$ -2x^2 + 2x + 40 $$ :::: ::::{solution} $$ -2(x - 5)(x + 4) = -2(x^2 + 4x - 5x - 20) = -2(x^2 - x -20) = -2x^2 + 2x + 40 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ -(x + 1)(x - 4) $$ ::::{answer} $$ -x^2 + 3x + 4 $$ :::: ::::{solution} $$ -(x + 1)(x - 4) = -(x^2 - 4x + x - 4) = -(x^2 - 3x - 4) = -x^2 + 3x + 4 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 11 --- level: 3 --- Nedenfor finner du en quiz om kvadratsetningene. Velg riktig alternativ som hører til uttrykket som er oppgitt. Du skal enten faktorisere eller utvide. Svar riktig på så mange som mulig før tiden går ut! :::{timed-quiz} Q: $$(x + 3)^2$$ + $$x^2 + 6x + 9$$ - $$x^2 + 3x + 9$$ - $$x^2 - 6x + 9$$ - $$x^2 + 3x - 9$$ Q: $$x^2 - 4x + 4$$ + $$(x - 2)^2$$ - $$(x + 2)^2$$ - $$-(x - 2)^2$$ - $$(x + 4)^2$$ Q: $$(x - 5)^2$$ + $$x^2 - 10x + 25$$ - $$x^2 + 10x + 25$$ - $$x^2 - 5x + 25$$ - $$x^2 - 25x + 5$$ Q: $$x^2 + 14x + 49$$ + $$(x + 7)^2$$ - $$(x - 7)^2$$ - $$(x + 14)^2$$ - $$(x + 49)^2$$ Q: $$(x - 2)(x + 2)$$ + $$x^2 - 4$$ - $$x^2 + 4$$ - $$x^2 - 2x + 4$$ - $$x^2 + 2x - 4$$ Q: $$4x^2 - 12x + 9$$ + $$(2x - 3)^2$$ - $$(2x + 3)^2$$ - $$(4x - 3)^2$$ - $$(2x - 9)^2$$ Q: $$(x + 4)^2$$ + $$x^2 + 8x + 16$$ - $$x^2 + 4x + 16$$ - $$x^2 - 8x + 16$$ - $$x^2 + 16x + 4$$ Q: $$x^2 - 36$$ + $$(x - 6)(x + 6)$$ - $$(x - 3)(x + 12)$$ - $$(x - 18)(x + 2)$$ - $$(x - 6)^2$$ Q: $$(2x + 5)^2$$ + $$4x^2 + 20x + 25$$ - $$2x^2 + 25x + 5$$ - $$4x^2 + 10x + 25$$ - $$2x^2 + 5x + 25$$ Q: $$x^2 + 2x + 1$$ + $$(x + 1)^2$$ - $$(x - 1)^2$$ - $$(x + 2)^2$$ - $$(x + 1)(x - 1)$$ Q: $$(x - 8)^2$$ + $$x^2 - 16x + 64$$ - $$x^2 + 16x + 64$$ - $$x^2 - 8x + 64$$ - $$x^2 - 64x + 8$$ Q: $$9x^2 - 1$$ + $$(3x - 1)(3x + 1)$$ - $$(9x - 1)(x + 1)$$ - $$(3x - 1)^2$$ - $$(x - 1)(x + 9)$$ Q: $$(x + 6)^2$$ + $$x^2 + 12x + 36$$ - $$x^2 + 6x + 36$$ - $$x^2 - 12x + 36$$ - $$x^2 + 36x + 6$$ Q: $$(2x - 7)^2$$ + $$4x^2 - 28x + 49$$ - $$2x^2 - 14x + 49$$ - $$4x^2 - 14x + 49$$ - $$4x^2 - 7x + 49$$ Q: $$(x + 8)^2$$ + $$x^2 + 16x + 64$$ - $$x^2 + 8x + 64$$ - $$x^2 - 16x + 64$$ - $$x^2 + 64x + 8$$ ::: :::::::::::::::