Множење полинома 2

Текст задатака објашњених у видео лекцији:

Пр.1)   Израчунати следеће производе:

           а) $\left( {3x - 8} \right) \cdot \left( {5x - 2} \right)$

           б) $\left( {a - 1} \right) \cdot \left( {5a + 4} \right)$

           в) $\left( { - \frac{1}{2}{x^2} + \frac{1}{4}x} \right) \cdot \left( {\frac{3}{4}{x^2} + 0,5x} \right)$

Пр.2)   Израчунати следеће производе:

           а) $\left( {x - 5} \right) \cdot \left( {{x^2} + 4x - 9} \right)$

           б) $\left( {2{y^2} + 4y - 3} \right) \cdot \left( {7y - 1} \right)$

           в) $\left( {{b^2} + b - 1} \right) \cdot \left( {4{b^2} - 3b - 2} \right)$

Пр.3)   Упростити изразе:

           а) $\left( {x - 5} \right) \cdot \left( {3{x^2} + x - 10} \right) - x \cdot \left( {5{x^2} + 2x - 11} \right)$

           б) $3{y^2} \cdot \left( {4{y^2} + 2y + 5} \right) - \left( {2y + 3} \right) \cdot \left( {6{y^3} - {y^2}} \right)$

           в) $\left( {x - 7} \right) \cdot \left( {5x - 8} \right) - \left( {3x + 8} \right) \cdot \left( {5x - 2} \right)$

Пр.4)   За колико је вредност израза $\left( {2x - 1} \right) \cdot \left( {2x - 2} \right)$

            већа од израза $\left( {4x + 2} \right) \cdot \left( {x - 2} \right)$?

Пр.1)   

а) 

\[\begin{gathered}
\left( {3x - 8} \right) \cdot \left( {5x - 2} \right) = 5x \cdot 3x - 2 \cdot 3x + 5x \cdot \left( { - 8} \right) - 2 \cdot \left( { - 8} \right) = \hfill \\
= 15{x^2} - 6x - 40x + 16 = 15{x^2} - 46x + 16 \hfill \\
\hfill \\
\end{gathered} \]

б) \[\left( {a - 1} \right) \cdot \left( {5a + 4} \right) = 5{a^2} + 4a - 5a - 4 = 5{a^2} - a - 4\]

в) 

\[\begin{gathered}
\left( { - \frac{1}{2}{x^2} + \frac{1}{4}x} \right) \cdot \left( {\frac{3}{4}{x^2} + 0,5x} \right) = \hfill \\
= - \frac{1}{2}{x^2} \cdot \frac{3}{4}{x^2} + \left( { - \frac{1}{2}{x^2}} \right) \cdot 0,5x + \frac{1}{4}x \cdot \frac{3}{4}{x^2} + \frac{1}{4}x \cdot 0,5x = \hfill \\
= - \frac{3}{8}{x^4} - \frac{1}{4}{x^3} + \frac{3}{{16}}{x^3} + \frac{1}{8}{x^2} = - \frac{3}{8}{x^4} - \frac{1}{{16}}{x^3} + \frac{1}{8}{x^2} \hfill \\
\end{gathered} \]

Пр.2)   

а) 

\[\begin{gathered}
\left( {x - 5} \right) \cdot \left( {{x^2} + 4x - 9} \right) = x \cdot {x^2} + x \cdot 4x - x \cdot 9 + {x^2} \cdot \left( { - 5} \right) + 4x \cdot \left( { - 5} \right) - 9 \cdot \left( { - 5} \right) = \hfill \\
= {x^3} + 4{x^2} - 9x - 5{x^2} - 20x + 45 = {x^3} - {x^2} - 29x + 45 \hfill \\
\end{gathered} \]

б) 

\[\begin{gathered}
\left( {2{y^2} + 4y - 3} \right) \cdot \left( {7y - 1} \right) = 14{y^3} - 2{y^2} + 28{y^2} - 4y - 21y + 3 = \hfill \\
= 14{y^3} - 26{y^2} - 25y + 3 \hfill \\
\end{gathered} \]

в) 

\[\begin{gathered}
\left( {{b^2} + b - 1} \right) \cdot \left( {4{b^2} - 3b - 2} \right) = 4{b^4} - 3{b^3} - 2{b^2} + 4{b^3} - 3{b^2} - 2b - 4{b^2} + 3b + 2 = \hfill \\
= 4{b^4} + {b^3} - 9{b^2} + b + 2 \hfill \\
\end{gathered} \]

Пр.3)   

а) 

\[\begin{gathered}
\left( {x - 5} \right) \cdot \left( {3{x^2} + x - 10} \right) - x \cdot \left( {5{x^2} + 2x - 11} \right) = \hfill \\
= 3{x^3} + {x^2} - 10x - 15{x^2} - 5x + 50 - 5{x^3} - 2{x^2} + 11x = \hfill \\
= - 2{x^3} - 16{x^2} - 4x + 50 \hfill \\
\end{gathered} \]

б) 

\[\begin{gathered}
3{y^2} \cdot \left( {4{y^2} + 2y + 5} \right) - \left( {2y + 3} \right) \cdot \left( {6{y^3} - {y^2}} \right) = \hfill \\
= 12{y^4} + 6{y^3} + 15{y^2} - \left( {12{y^4} - 2{y^3} + 18{y^3} - 3{y^2}} \right) = \hfill \\
= 12{y^4} + 6{y^3} + 15{y^2} - 12{y^4} + 2{y^3} - 18{y^3} + 3{y^2} = \hfill \\
= - 10{y^3} + 18{y^2} \hfill \\
\end{gathered} \]

в) 

\[\begin{gathered}
\left( {x - 7} \right) \cdot \left( {5x - 8} \right) - \left( {3x + 8} \right) \cdot \left( {5x - 2} \right) = \hfill \\
= 5{x^2} - 8x - 35x + 56 - \left( {15{x^2} - 6x + 40x - 16} \right) = \hfill \\
= - 10{x^2} - 77x + 72 \hfill \\
\end{gathered} \]

Пр.4)   

\[\begin{gathered}
\left( {2x - 1} \right) \cdot \left( {2x - 2} \right) - \left( {4x + 2} \right) \cdot \left( {x - 2} \right) = \hfill \\
= 4{x^2} - 4x - 2x + 2 - \left( {4{x^2} - 8x + 2x - 4} \right) = \hfill \\
= 4{x^2} - 4x - 2x + 2 - 4{x^2} + 8x - 2x + 4 = 6 \hfill \\
\end{gathered} \]