Sievennä lausekkeet
a) $(x+7)(x-7)$
b) $(2x+3)^2$
c) $(x^2-5)^2$
Hyödynnetään ratkaisuissa muistikaavoja.
\(\begin{align*} \textbf{a) } \quad &(x+7)(x-7) &&||\ (a+b)(a-b)=a^2-b^2 \\ =&x^2- 7^2 &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{x^2-49}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) } \quad &(2x+3)^2 &&||\ (a+b)^2=a^2+2ab+b^2 \\ =&(2x)^2+ 2 \cdot 2x \cdot 3+3^2 &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{4x^2+12x+9}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } \quad &(x^2-5)^2 &&||\ (a-b)^2=a^2-2ab+b^2 \\ =&(x^2)^2- 2 \cdot x^2 \cdot 5+5^2 &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{x^4-10x^2+25}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Jaa polynomi tekijöihin.
a) $7a-7b+14c$
b) $2x^3-50x$
c) $x^2-25$
\(\begin{align*} \textbf{a) } \quad &7a-7b+14c&&||\ \text{yhteisenä tekijänä luku 7} \\ =&7 \cdot a - 7 \cdot b + 7 \cdot 2c &&\require{color}\color{red}{\text{(+1p)}}\\=&\underline{\underline{7(a-b+2c)}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) } \quad &2x^3-50x &&||\ \text{yhteisenä tekijänä }2x \\ =&2x \cdot x^2- 2x \cdot 25 &&\require{color}\color{red}{\text{(+1p)}} \\ =&2x(x^2-25)\\ =&\underline{\underline{2x(x+5)(x-5)}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } \quad &x^2-25 &&||\ a^2-b^2=(a+b)(a-b) \\ =&x^2- 5^2 &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{(x+5)(x-5)}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Jaa polynomi tekijöihin.
a) $x^2+\frac12x+\frac{1}{16}$
b) $x^2-8x+16$
c) $2xy-4x+4y-8$
\(\begin{align*} \textbf{a) } \quad &x^2+\frac12x+\frac{1}{16} &&||\ a^2+2ab+b^2=(a+b)^2 \\ =&x^2 +2 \cdot x \cdot \frac14 +\Big(\frac14\Big)^2 &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{\Big(x+\frac14\Big)^2}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) } \quad &x^2-8x+16 &&||\ a^2-2ab+b^2=(a-b)^2 \\ =&x^2 -2 \cdot x \cdot4 +4^2 &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{(x-4)^2}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } \quad &2xy-4x+4y-8 &&||\ \text{ryhmittely} \\ =&2x(y-2)+4(y-2) &&\require{color}\color{red}{\text{(+1p)}} \\ =&\underline{\underline{(2x+4)(y-2)}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Tekemäsi itsearvion pohjalta tuloksesi prosentteina on: