MAA2 testi1.1

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Suositeltava osaamistaso: 
90%

Täydennä taulukkoon kohdat a)-f).

\(\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|c|c|} \hline \text{Polynomi} & \begin{array}{c} \text{Termien} \\ \text{lukumäärä} \end{array} & \begin{array}{c}\text{Korkeimman} \\ \text{asteen} \\ \text{termin} \\ \text{kerroin} \end{array} & \text{Asteluku} & \text{Vakiotermi}\T \\ \hline -3x^{2}+5x & 2 \phantom{\Big|} & -3 & 2 & 0 \\ \hline 9x^{4}-x-7 & 3 \phantom{\Big|} & \text{a)} & \text{b)} & \text{c)} \\ \hline \text{d)} & 3 \phantom{\Big|} & 8 & 2 & 5 \\ \hline \text{e)} & 1 \phantom{\Big|} & -7 & 67 & \text{f)} \\ \hline \end{array}\)

Pisteytysohje: 

\(\textbf{a) } 9 \qquad\quad \require{color}\color{red}{\text{(+1p)}}\)

\(\textbf{b) } 4 \qquad\quad \require{color}\color{red}{\text{(+1p)}}\)

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

\(\begin{align*}\textbf{d) } &\text{Esimerkiksi polynomi } \\ &8x^2+x+5 \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\textbf{e) } -7x^{67} \quad \require{color}\color{red}{\text{(+1p)}}\)

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

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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 \(x=-2\)).

Pisteytysohje: 

\(\begin{align*} \textbf{a) }& \quad P(x)+Q(x) \\ &=(-3x^2+5x)+(5x^3-x^2-x+19) \quad &&\require{color}\color{red}{\text{(+1p)}} \\ &=-3x^2+5x+5x^3-x^2-x+19 \\ &=\underline{\underline{5x^3-4x^2+4x+19}} && \color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{b) } & \quad P(x)-Q(x) \\ & =(-3x^2+5x)-(5x^3-x^2-x+19) \quad && \require{color}\color{red}{\text{(+1p)}} \\ &=-3x^2+5x-5x^3+x^2+x-19 \\ &=\underline{\underline{-5x^3-2x^2+6x-19} }\quad &&\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{c) } Q(-2)&=5 \cdot (-2)^3 - (-2)^2 -(-2)+19&& \require{color}\color{red}{\text{(+1p)}} \\ &=5 \cdot (-8) - 4 +2 + 19 \\ &=-40+17 \\ &=\underline{\underline{-23}} \quad&& \color{red}{\text{(+1p)}} \end{align*}\)

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Sievennä lausekkeet 
a) \(3(2a+b)+a-3b\) 

b) $ab+a+b+2ab$ 

c) $-2(x-2)-3(-3x+3y)$

d) $-4x^{4} \cdot 3x^3$

e) $-8xy \cdot 2x^2y^3 \cdot (-2x^3y^5)$

f) $(x-3)(y-2)$ 

Pisteytysohje: 

\( \begin{align*} \textbf{a) }& \quad 3(2a+b)+a-3b \\ &=6a+3b+a-3b \\ &=\underline{\underline{ \ 7a \ }} && \require{color}\color{red}{\text{(+1p)}} \end{align*} \)

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

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

\(\begin{align*} \textbf{d) } \quad &-4x^4 \cdot 3x^3 \\ =&-4 \cdot 3 \cdot x^4 \cdot x^3 \\ =&-12 x^{4+3} \\ =&\underline{\underline{-12x^7}} ​ \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)

\(\begin{align*} \textbf{e) } \quad &-8xy \cdot 2x^2y^3 \cdot (-2x^3y^5) \\ =&-8 \cdot 2 \cdot x \cdot y \cdot x^2 \cdot y^3 (-2x^3y^5) \\ =&-16x^3y^4 \cdot (-2x^3y^5) \\ =&\underline{\underline{32x^6y^9}} \quad \require{color}\color{red}{\text{(+1p)}} \\ \end{align*}\)

\(\begin{align*} \textbf{f) } \quad &(x-3)(y-2) \\ =& x \cdot y + x \cdot (-2) -3 \cdot y -3 \cdot (-2)\\ =&xy-2x-3y+6 \\ =&\underline{\underline{-2x+xy-3y+6 }} \quad \require{color}\color{red}{\text{(+1p)}} \\ \end{align*}\)

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