Question 1177973
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The answer can be obtained with FAR less work than shown by the other tutor....<br>
{{{(x^3-x^2-5x-2)(x^4+x^3+kx^2-5x+2)=x^7-4x^5-14x^4-5x^3+19x^2-4}}}<br>
In the equation, k is the coefficient of the x^2 term.  So look at ANY one coefficient in the product of a term which is x^2 or a higher power of x.<br>
Example: The coefficient of the x^2 term in the product is 19.  That coefficient comes from (1) the constant term in the first expression times the x^2 term in the second, (2) the product of the linear terms in the two expressions, and (3) the x^2 term in the first expression and the constant term in the second:<br>
{{{19 = (-1)(2)+(-5)(-5)+(-2)(k)}}}
{{{19 = -2+25-2k}}}
{{{2k = 4}}}
{{{k = 2}}}<br>
ANSWER: k=2<br>
I could have used any other coefficient of the product of a term with degree 2 or higher.<br>
Second example, using the coefficient of the x^4 term in the product:<br>
{{{-14 = (1)(-5)+(-1)(k)+(-5)(1)+(-2)(1)}}}
{{{-14 = -5-k-5-2}}}
{{{k = 2}}}<br>