Täydennä taulukkoon kohdat a)-f).
\(\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|c|c|} \hline \text{Polynomi} & \begin{array}{c} \text{Termien} \\ \text{lukumäärä} \end{array} & \begin{array}{c}\text{Korkeimman} \\ \text{asteen} \\ \text{termin} \\ \text{kerroin} \end{array} & \text{Asteluku} & \text{Vakiotermi}\T \\ \hline 4x^{2}-7x+5 & 3 \phantom{\Big|} & 4 & 2 & 5 \\ \hline -11x^{5}+x & 2 \phantom{\Big|} & \text{a)} & \text{b)} & \text{c)} \\ \hline \text{d)} & 2 \phantom{\Big|} & -6 & 3 & -13 \\ \hline \text{e)} & 2 \phantom{\Big|} & 7 & 22 & \text{f)} \\ \hline \end{array}\)
\(\textbf{a) } -11 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{b) } 5 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\textbf{c) } 0 \quad \require{color}\color{red}{\text{(+1p)}}\)
\(\begin{align*}\textbf{d) } &\text{Esimerkiksi polynomi } \\ &-6x^3-13 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*}\textbf{e) }&\text{Esimerkiksi polynomi} \\ &7x^{22}+1 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\textbf{f) } \text{Katso e) -kohta, } 1 \quad \require{color}\color{red}{\text{(+1p)}}\)
Olkoon polynomit \(P(x)=3x^2-4x+5\) ja \(Q(x)=-2x^4+4x^2-x-11\).
a) Laske polynomien summa \(P(x)+Q(x)\)
b) Laske polynomien erotus \(P(x)-Q(x)\)
c) Laske \(Q(-1)\) (eli polynomin \(Q(x)\) arvo, kun \(x=-1\)).
\(\begin{align*} \textbf{a) } & \quad P(x)+Q(x)\\ &=(3x^2-4x+5)+(-2x^4+4x^2-x-11) \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=3x^2-4x-2x^4+4x^2-x-11 \\ &=\underline{\underline{-2x^4+7x^2-5x-11}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{b) } & \quad P(x)-Q(x) \\ &=(3x^2-4x+5)-(-2x^4+4x^2-x-11) \quad \require{color}&&\color{red}{\text{(+1p)}} \\ &=3x^2-4x+5+2x^4-4x^2+x+11 \\ &=\underline{\underline{2x^4+x^2-3x+16}} && \color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } Q(-1)&=-2 \cdot (-1)^4 + 4 \cdot (-1)^2 -(-1)-11 \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=-2 \cdot 1 +4 \cdot 1 + 1 -11 \\ &=-2+4+1-11 \\ &=\underline{\underline{\ -8 \ }} &&\require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Sievennä lausekkeet
a) \(3(2a+b)+a-3b\)
b) $-2(x-2)-3(-3x+3y)$
c) $-4x^{4} \cdot 3x^3$
\( \begin{align*} \textbf{a) }& \quad 3(2a+b)+a-3b \\ &=6a+3b+a-3b && \require{color}\color{red}{\text{(+1p)}} \\ &=\underline{\underline{ \ 7a \ }} && \require{color}\color{red}{\text{(+1p)}} \end{align*} \)
\(\begin{align*} \textbf{b) }\quad &-2(x-2)-3(-3x+3y) \\ =&-2 \cdot x -2 \cdot (-2) -3 \cdot (-3x) -3 \cdot 3y \ \ &&\require{color}\color{red}{\text{(+1p)}} \\ =&-2x+4+9x-9y \\ =&\underline{\underline{7x-9y+4}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{c) } \quad & -4x^4 \cdot 3x^3 \\ =& -4 \cdot 3 \cdot x^4 \cdot x^3 && \require{color}\color{red}{\text{(+1p)}} \\ =& -12 \cdot x^{4+3} \\ =&-12 x^7 && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
Tekemäsi itsearvion pohjalta tuloksesi prosentteina on: