Täydennä taulukkoon kohdat a)-f).
\(\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|c|c|} \hline \text{Polynomi} & \begin{array}{c} \text{Termien} \\ \text{lukumäärä} \end{array} & \begin{array}{c}\text{Korkeimman} \\ \text{asteen} \\ \text{termin} \\ \text{kerroin} \end{array} & \text{Asteluku} & \text{Vakiotermi}\T \\ \hline -3x^{2}+5x & 2 \phantom{\Big|} & -3 & 2 & 0 \\ \hline 9x^{4}-x-7 & 3 \phantom{\Big|} & \text{a)} & \text{b)} & \text{c)} \\ \hline \text{d)} & 3 \phantom{\Big|} & 8 & 2 & 5 \\ \hline \text{e)} & 1 \phantom{\Big|} & -7 & 67 & \text{f)} \\ \hline \end{array}\)
\(\textbf{a) } 9 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{b) } 4 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{c) } -7 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\begin{align*}\textbf{d) } &\text{Esimerkiksi polynomi } \\ &8x^2+x+5 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\textbf{e) } -7x^{67} \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{f) } 0 \quad \require{color}\color{red}{\text{(+1p)}}\)
Olkoon polynomit \(P(x)=-3x^{2} +5x\) ja \(Q(x)=5x^{3}-x^{2}-x+19\).
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 \(x=-2\)).
\(\begin{align*} \textbf{a) }& \quad P(x)+Q(x) \\ &=(-3x^2+5x)+(5x^3-x^2-x+19) \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=-3x^2+5x+5x^3-x^2-x+19 \\ &=\underline{\underline{5x^3-4x^2+4x+19}} && \color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) } & \quad P(x)-Q(x) \\ & =(-3x^2+5x)-(5x^3-x^2-x+19) \quad && \require{color}\color{red}{\text{(+1p)}} \\ &=-3x^2+5x-5x^3+x^2+x-19 \\ &=\underline{\underline{-5x^3-2x^2+6x-19} }\quad &&\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } Q(-2)&=5 \cdot (-2)^3 - (-2)^2 -(-2)+19&& \require{color}\color{red}{\text{(+1p)}} \\ &=5 \cdot (-8) - 4 +2 + 19 \\ &=-40+17 \\ &=\underline{\underline{-23}} \quad&& \color{red}{\text{(+1p)}} \end{align*}\)
Sievennä lausekkeet
a) \(4(a+b)+2b \)
b) \(2(x-3)+3(-2y+2)\)
c) \(6x^{2}(-5x^3)\)
\(\begin{align*} \textbf{a) } \quad &4(a+b)+2b \\ =&4\cdot a + 4 \cdot b + 2 b \quad &&\require{color}\color{red}{\text{(+1p)}} \\ =&4a+4b+2b \\ =&\underline{\underline{4a+6b}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) }& \ \ \ \ 2(x-3)+3(-2y+2) \\ &= 2 \cdot x+ 2 \cdot (-3) + 3 \cdot (-2y) + 3 \cdot 2&& \require{color}\color{red}{\text{(+1p)}} \\ &=2x-6-6y+6 \\ &=\underline{\underline{2x-6y}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } \quad &6x^2(-5x^3) \\ =&6 \cdot (-5) \cdot x^2 \cdot x^3 &&\require{color}\color{red}{\text{(+1p)}} \\ =&-30 x^{2+3} \\ =&\underline{\underline{-30x^5}} &&\color{red}{\text{(+1p)}} \end{align*}\)
Tekemäsi itsearvion pohjalta tuloksesi prosentteina on: