MAA2 testi1.3

Testin aihepiiri: 
Polynomien summa, erotus ja tulo
Suositeltava osaamistaso: 
90%

Täydennä taulukkoon kohdat a)-f).

\(\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|c|c|} \hline \text{Polynomi} & \begin{array}{c} \text{Termien} \\ \text{lukumäärä} \end{array} & \begin{array}{c}\text{Korkeimman} \\ \text{asteen} \\ \text{termin} \\ \text{kerroin} \end{array} & \text{Asteluku} & \text{Vakiotermi}\T \\ \hline-8x^7+x^3 & 2 \phantom{\Big|} & -8 & 7 & 0 \\ \hline -x^6-x^3-7& 3 \phantom{\Big|} & \text{a)} & \text{b)} & \text{c)} \\ \hline \text{d)} & 2 \phantom{\Big|} & 1 & 1 & 1 \\ \hline \text{e)} & 1 \phantom{\Big|} & -1 & f) &0 \\ \hline \end{array}\)

Pisteytysohje: 

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

\(\textbf{b) } 6 \quad \require{color}\color{red}{\text{(+1p)}}\)

\(\textbf{c) } -7 \quad \require{color}\color{red}{\text{(+1p)}}\)

\(\begin{align*}\textbf{d) } &\text{Polynomi } \\ &x+1 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*}\textbf{e) }&\text{Esimerkiksi polynomi}\\& -x^{67}\require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*}\textbf{f) }& \text{e) - kohdan perusteella} \\ & 67 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

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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 \(x=-2\)).

Pisteytysohje: 

\(\begin{align*} \textbf{a) }& \quad P(x)+Q(x) \\ &=(4x^2+12x+2)+(-x^3-2x^2-3x) \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=4x^2+12x+2-x^3-2x^2-3x \\ &=\underline{\underline{-x^3+2x^2+9x+2}} && \color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{b) } & \quad P(x)-Q(x) \\ & =(4x^2+12x+2)-(-x^3-2x^2-3x) \quad && \require{color}\color{red}{\text{(+1p)}} \\ &=4x^2+12x+2+x^3+2x^2+3x \\ &=\underline{\underline{x^3+6x^2+15x+2} }\quad &&\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{c) } Q(-2)&=- (-2)^3 -2 \cdot (-2)^2 -3 \cdot (-2)&& \require{color}\color{red}{\text{(+1p)}} \\ &=- (-8) - 2 \cdot4 + 6 \\ &=8-8+6 \\ &=\underline{\underline{\ 6\ }} \quad&& \color{red}{\text{(+1p)}} \end{align*}\)

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Sievennä lausekkeet 
a) \(3a-4(a-b)\)

b) $-3ab+7a+9b+2ab$ 

c) \(5(-y+2)-(-x-6y)\)

d) $9x^{6} \cdot 2x^7$

e) \(-x^5(-7x^6)\) 

f) $(2x+4)(2y-4)$ 

Pisteytysohje: 

\( \begin{align*} \textbf{a) }& \quad 3a-4(a-b) \\ &=3a-4a+4b \\ &=\underline{\underline{ -a+4b }} && \require{color}\color{red}{\text{(+1p)}} \end{align*} \)

\(\begin{align*} \textbf{b) }\quad &-3ab+7a+9b+2ab \\ =&\underline{\underline{7a-ab+9b}} \ \ && \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

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

\(\begin{align*} \textbf{d) } \quad &9x^6 \cdot 2x^7 \\ =&9 \cdot 2 \cdot x^6 \cdot x^7 \\ =&18 x^{6+7} \\ =&\underline{\underline{18x^{13}}} ​ \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{e)} \quad &-x^5(-7x^6) \\ =&-1 \cdot (-7) \cdot x^5 \cdot x^6 \\ =&7x^{11}&& \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{f) } \quad &(2x+4)(2y-4) \\ =& 2x \cdot2 y +2 x \cdot (-4) +4 \cdot 2y +4 \cdot (-4)\\ =&4xy-8x+8y-16 \\ =&\underline{\underline{-8x+4xy+8y-16 }} \quad \require{color}\color{red}{\text{(+1p)}} \\ \end{align*}\)

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