MAA2 testi1.2

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Suositeltava osaamistaso: 
90%

Täydennä taulukkoon kohdat a)-f).

\(\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|c|c|} \hline \text{Polynomi} & \begin{array}{c} \text{Termien} \\ \text{lukumäärä} \end{array} & \begin{array}{c}\text{Korkeimman} \\ \text{asteen} \\ \text{termin} \\ \text{kerroin} \end{array} & \text{Asteluku} & \text{Vakiotermi}\T \\ \hline 4x^{2}-7x+5 & 3 \phantom{\Big|} & 4 & 2 & 5 \\ \hline -11x^{5}+x & 2 \phantom{\Big|} & \text{a)} & \text{b)} & \text{c)} \\ \hline \text{d)} & 2 \phantom{\Big|} & -6 & 3 & -13 \\ \hline \text{e)} & 2 \phantom{\Big|} & 7 & 22 & \text{f)} \\ \hline \end{array}\)

Pisteytysohje: 

\(\textbf{a) } -11 \quad \require{color}\color{red}{\text{(+1p)}}\)

\(\textbf{b) } 5 \quad \require{color}\color{red}{\text{(+1p)}}\)

\(\textbf{c) } 0 \quad \require{color}\color{red}{\text{(+1p)}}\)

\(\begin{align*}\textbf{d) } &\text{Esimerkiksi polynomi } \\ &-6x^3-13 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*}\textbf{e) }&\text{Esimerkiksi polynomi} \\ &7x^{22}+1 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\textbf{f) } \text{Katso e) -kohta, } 1 \quad \require{color}\color{red}{\text{(+1p)}}\)

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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 \(x=-1\)).

Pisteytysohje: 

\(\begin{align*} \textbf{a) } & \quad P(x)+Q(x)\\ &=(3x^2-4x+5)+(-2x^4+4x^2-x-11) \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=3x^2-4x+5-2x^4+4x^2-x-11 \\ &=\underline{\underline{-2x^4+7x^2-5x-6}} && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{b) } & \quad P(x)-Q(x) \\ &=(3x^2-4x+5)-(-2x^4+4x^2-x-11) \quad \require{color}&&\color{red}{\text{(+1p)}} \\ &=3x^2-4x+5+2x^4-4x^2+x+11 \\ &=\underline{\underline{2x^4-x^2-3x+16}} && \color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{c) } Q(-1)&=-2 \cdot (-1)^4 + 4 \cdot (-1)^2 -(-1)-11 \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=-2 \cdot 1 +4 \cdot 1 + 1 -11 \\ &=-2+4+1-11 \\ &=\underline{\underline{\ -8 \ }} &&\require{color}\color{red}{\text{(+1p)}} \end{align*}\)

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Sievennä lausekkeet 
a) $2(a-2b)+2a-2b$

b) $ab-a-3b-2ab$ 

c) $-4(2x-3)+2(-x+y)$

d) $5x^{3} \cdot (-2x^3)$

e) $-3x^2y \cdot 3x^3y^2 \cdot (-5xy^4)$

f) $(x-1)(y+5)$ 

Pisteytysohje: 

\( \begin{align*} \textbf{a) }& \quad 2(a-2b)+2a-2b \\ &=2a-4b+2a-2b \\ &=\underline{\underline{ \ 4a-6b \ }} && \require{color}\color{red}{\text{(+1p)}} \end{align*} \)

\(\begin{align*} \textbf{b) }\quad &ab-a-3b-2ab \\ =&\underline{\underline{-a-ab-3b}} \ \ && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{c) } \quad &-4(2x-3)+2(-x+y) \\ =&-4 \cdot 2x -4 \cdot (-3) +2 \cdot (-x)+2 \cdot y \\ =&-8x+12-2x+2y \\ =&\underline{\underline{-10x+2y+12}} \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{d) } \quad &5x^{3} \cdot (-2x^3) \\ =&5 \cdot (-2) \cdot x^3 \cdot x^3 \\ =&-10 x^{3+3} \\ =&\underline{\underline{-10x^6}} ​ \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{e) } \quad &-3x^2y \cdot 3x^3y^2 \cdot (-5xy^4) \\ =&-3 \cdot 3 \cdot x^2 \cdot y \cdot x^3 \cdot y^2 \cdot (-5xy^4) \\ =&-9x^5y^3 \cdot (-5xy^4) \\ =&\underline{\underline{45x^6y^7}} \quad \require{color}\color{red}{\text{(+1p)}} \\ \end{align*}\)

\(\begin{align*} \textbf{f) } \quad &(x-1)(y+5) \\ =& x \cdot y + x \cdot 5 -1 \cdot y -3 \cdot 5\\ =&xy+5x-y-5 \\ =&\underline{\underline{5x+xy-y-5 }} \quad \require{color}\color{red}{\text{(+1p)}} \\ \end{align*}\)

 

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