Täydennä taulukkoon kohdat a)-f).
\(\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|c|c|} \hline \text{Polynomi} & \begin{array}{c} \text{Termien} \\ \text{lukumäärä} \end{array} & \begin{array}{c}\text{Korkeimman} \\ \text{asteen} \\ \text{termin} \\ \text{kerroin} \end{array} & \text{Asteluku} & \text{Vakiotermi}\T \\ \hline-8x^7+x^3 & 2 \phantom{\Big|} & -8 & 7 & 0 \\ \hline -x^6-x^3-7& 3 \phantom{\Big|} & \text{a)} & \text{b)} & \text{c)} \\ \hline \text{d)} & 2 \phantom{\Big|} & 1 & 1 & 1 \\ \hline \text{e)} & 1 \phantom{\Big|} & -1 & f) &0 \\ \hline \end{array}\)
\(\textbf{a) } -1 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{b) } 6 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{c) } -7 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\begin{align*}\textbf{d) } &\text{Polynomi } \\ &x+1 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*}\textbf{e) }&\text{Esimerkiksi polynomi}\\& -x^{67}\require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*}\textbf{f) }& \text{e) - kohdan perusteella} \\ & 67 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Olkoon polynomit \(P(x)=4x^2+12x+2\) ja \(Q(x)=-x^3-2x^2-3x\).
a) Laske polynomien summa \(P(x)+Q(x)\)
b) Laske polynomien erotus \(P(x)-Q(x)\)
c) Laske \(Q(-2)\) (eli polynomin \(Q(x)\) arvo, kun \(x=-2\)).
\(\begin{align*} \textbf{a) }& \quad P(x)+Q(x) \\ &=(4x^2+12x+2)+(-x^3-2x^2-3x) \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=4x^2+12x+2-x^3-2x^2-3x \\ &=\underline{\underline{-x^3+2x^2+9x+2}} && \color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) } & \quad P(x)-Q(x) \\ & =(4x^2+12x+2)-(-x^3-2x^2-3x) \quad && \require{color}\color{red}{\text{(+1p)}} \\ &=4x^2+12x+2+x^3+2x^2+3x \\ &=\underline{\underline{x^3+6x^2+15x+2} }\quad &&\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } Q(-2)&=- (-2)^3 -2 \cdot (-2)^2 -3 \cdot (-2)&& \require{color}\color{red}{\text{(+1p)}} \\ &=- (-8) - 2 \cdot4 + 6 \\ &=8-8+6 \\ &=\underline{\underline{\ 6\ }} \quad&& \color{red}{\text{(+1p)}} \end{align*}\)
Sievennä lausekkeet
a) \(3a-4(a-b)\)
b) \(5(-y+2)-(-x-6y)\)
c) \(-x^5(-7x^6)\)
\( \begin{align*} \textbf{a) }& \quad 3a-4(a-b) \\ &=3a-4a+4b && \require{color}\color{red}{\text{(+1p)}} \\ &=\underline{\underline{ -a+4b }} && \require{color}\color{red}{\text{(+1p)}} \end{align*} \)
\(\begin{align*} \textbf{b) }\quad &5(-y+2)-(-x-6y) \\ =&5 \cdot (-y) + 5 \cdot 2 +x+6y\ \ &&\require{color}\color{red}{\text{(+1p)}} \\ =&-5y+10+x+6y \\ =&\underline{\underline{x+y+10}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c)} \quad &-x^5(-7x^6) \\ =&-1 \cdot (-7) \cdot x^5 \cdot x^6 && \require{color}\color{red}{\text{(+1p)}} \\ =&7x^{11}&& \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Tekemäsi itsearvion pohjalta tuloksesi prosentteina on: