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Factor \(\displaystyle - 90 x^{3} + 50 x^{2} - 81 x + 45\).

Factoring out the GCF \(\displaystyle -1\) from each term gives \(\displaystyle -(90 x^{3} - 50 x^{2} + 81 x - 45)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 10 x^{2}\), gives \(\displaystyle 10 x^{2}(9 x - 5)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 9\), gives \(\displaystyle 9(9 x - 5)\). The polynomial now has a common binomial factor of \(\displaystyle 9 x - 5\). This gives \(\displaystyle -1[10 x^{2} \left(9 x - 5\right) +9 \cdot \left(9 x - 5\right)] = -\left(9 x - 5\right) \left(10 x^{2} + 9\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 90 x^{3} + 50 x^{2} - 81 x + 45$. 
    \soln{9cm}{Factoring out the GCF $-1$ from each term gives $-(90 x^{3} - 50 x^{2} + 81 x - 45)$. Grouping the first two terms and factoring out their GCF, $10 x^{2}$, gives $10 x^{2}(9 x - 5)$. Grouping the last two terms and factoring out their GCF, $9$, gives $9(9 x - 5)$. The polynomial now has a common binomial factor of $9 x - 5$. This gives $-1[10 x^{2} \left(9 x - 5\right) +9 \cdot \left(9 x - 5\right)] = -\left(9 x - 5\right) \left(10 x^{2} + 9\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 90 x^{3} + 50 x^{2} - 81 x + 45 " src="/equation_images/%20%5Cdisplaystyle%20-%2090%20x%5E%7B3%7D%20%2B%2050%20x%5E%7B2%7D%20-%2081%20x%20%2B%2045%20" alt="LaTeX:  \displaystyle - 90 x^{3} + 50 x^{2} - 81 x + 45 " data-equation-content=" \displaystyle - 90 x^{3} + 50 x^{2} - 81 x + 45 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle -1 " src="/equation_images/%20%5Cdisplaystyle%20-1%20" alt="LaTeX:  \displaystyle -1 " data-equation-content=" \displaystyle -1 " />  from each term gives  <img class="equation_image" title=" \displaystyle -(90 x^{3} - 50 x^{2} + 81 x - 45) " src="/equation_images/%20%5Cdisplaystyle%20-%2890%20x%5E%7B3%7D%20-%2050%20x%5E%7B2%7D%20%2B%2081%20x%20-%2045%29%20" alt="LaTeX:  \displaystyle -(90 x^{3} - 50 x^{2} + 81 x - 45) " data-equation-content=" \displaystyle -(90 x^{3} - 50 x^{2} + 81 x - 45) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 10 x^{2} " src="/equation_images/%20%5Cdisplaystyle%2010%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 10 x^{2} " data-equation-content=" \displaystyle 10 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 10 x^{2}(9 x - 5) " src="/equation_images/%20%5Cdisplaystyle%2010%20x%5E%7B2%7D%289%20x%20-%205%29%20" alt="LaTeX:  \displaystyle 10 x^{2}(9 x - 5) " data-equation-content=" \displaystyle 10 x^{2}(9 x - 5) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 9 " src="/equation_images/%20%5Cdisplaystyle%209%20" alt="LaTeX:  \displaystyle 9 " data-equation-content=" \displaystyle 9 " /> , gives  <img class="equation_image" title=" \displaystyle 9(9 x - 5) " src="/equation_images/%20%5Cdisplaystyle%209%289%20x%20-%205%29%20" alt="LaTeX:  \displaystyle 9(9 x - 5) " data-equation-content=" \displaystyle 9(9 x - 5) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 9 x - 5 " src="/equation_images/%20%5Cdisplaystyle%209%20x%20-%205%20" alt="LaTeX:  \displaystyle 9 x - 5 " data-equation-content=" \displaystyle 9 x - 5 " /> . This gives  <img class="equation_image" title=" \displaystyle -1[10 x^{2} \left(9 x - 5\right) +9 \cdot \left(9 x - 5\right)] = -\left(9 x - 5\right) \left(10 x^{2} + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20-1%5B10%20x%5E%7B2%7D%20%5Cleft%289%20x%20-%205%5Cright%29%20%2B9%20%5Ccdot%20%5Cleft%289%20x%20-%205%5Cright%29%5D%20%3D%20-%5Cleft%289%20x%20-%205%5Cright%29%20%5Cleft%2810%20x%5E%7B2%7D%20%2B%209%5Cright%29%20" alt="LaTeX:  \displaystyle -1[10 x^{2} \left(9 x - 5\right) +9 \cdot \left(9 x - 5\right)] = -\left(9 x - 5\right) \left(10 x^{2} + 9\right) " data-equation-content=" \displaystyle -1[10 x^{2} \left(9 x - 5\right) +9 \cdot \left(9 x - 5\right)] = -\left(9 x - 5\right) \left(10 x^{2} + 9\right) " /> . </p> </p>