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Find t if the expansion of the product of x^3 - 4x^2 + 2x - 5 and x^2 + tx - 7 has no x^2 term.

Guest Dec 4, 2014

Best Answer 

 #1
avatar+19653 
+10

Find t if the expansion of the product of x^3 - 4x^2 + 2x - 5 and x^2 + tx - 7 has no x^2 term.

$$\small{\text{
$
(x^3 - 4x^2 + 2x - 5)*( x^2 + tx - 7) = tx^4-4tx^3+\textcolor[rgb]{1,0,0}{2tx^2}-5tx+x^5-4x^4-5x^3+\textcolor[rgb]{1,0,0}{23x^2}-14x+35
$
}}$$

$$\\ \small{\text{
Set $
\textcolor[rgb]{1,0,0}{2tx^2}+\textcolor[rgb]{1,0,0}{23x^2} = 0
$
than the product has no $x^2$, t must be a constant!
}}
\\\\
2tx^2 + 23x^2 = 0 \\\\
2tx^2 = - 23x^2 \quad | \quad : 2x^2 \\\\
t= -\frac{23}{2} = - 11.5$$

$$\small{\text{
$
\textcolor[rgb]{1,0,0}{t= -11.5}\qquad (x^3-4x^2+2x-5)*(x^2+\textcolor[rgb]{1,0,0}{(-11.5)}x-7) = x^5 - 15.5x^4+41x^3+43.5x+35
$
}}$$

There is no more $$x^2$$

heureka  Dec 4, 2014
 #1
avatar+19653 
+10
Best Answer

Find t if the expansion of the product of x^3 - 4x^2 + 2x - 5 and x^2 + tx - 7 has no x^2 term.

$$\small{\text{
$
(x^3 - 4x^2 + 2x - 5)*( x^2 + tx - 7) = tx^4-4tx^3+\textcolor[rgb]{1,0,0}{2tx^2}-5tx+x^5-4x^4-5x^3+\textcolor[rgb]{1,0,0}{23x^2}-14x+35
$
}}$$

$$\\ \small{\text{
Set $
\textcolor[rgb]{1,0,0}{2tx^2}+\textcolor[rgb]{1,0,0}{23x^2} = 0
$
than the product has no $x^2$, t must be a constant!
}}
\\\\
2tx^2 + 23x^2 = 0 \\\\
2tx^2 = - 23x^2 \quad | \quad : 2x^2 \\\\
t= -\frac{23}{2} = - 11.5$$

$$\small{\text{
$
\textcolor[rgb]{1,0,0}{t= -11.5}\qquad (x^3-4x^2+2x-5)*(x^2+\textcolor[rgb]{1,0,0}{(-11.5)}x-7) = x^5 - 15.5x^4+41x^3+43.5x+35
$
}}$$

There is no more $$x^2$$

heureka  Dec 4, 2014
 #2
avatar+87333 
0

Nice, heureka......the solution is easy.....figuring out how to get there is the hard part....!!!!!

 

 

CPhill  Dec 4, 2014

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