Find t if the expansion of the product of 2x^3 + x^2 - 3x + 5 and x^4 - 7x^3 + tx^2 - 11x + 15 has no x^2 term.

 Apr 12, 2024

We try to pick out all the combinations of terms, one from each polynomial, such that their product is some number times \(x^2\).


First, if we pick 2x^3 from the first polynomial, it is impossible to get x^2 by multiplying it to another term from the other polynomial, we do not care about this case in this scenario.


Then, we consider the second term x^2 from the first polynomial. To get some multiple of x^2, we can only choose the constant term from the second polynomial, which will give us \(+15x^2\) if we multiply the terms.


Then, for -3x, we choose -11x from the second polynomial, so when we multiply them, we get \(+33x^2\)


For +5, we choose tx^2 from the second polynomial, so when we multiply them we get \(+5tx^2\).


Therefore, the term in x^2 is \(15x^2 + 33x^2 + 5tx^2\), which is \((5t + 48)x^2\). Since we are given that the product has no x^2 term, the coefficient must be 0, which gives \(t= -\dfrac{48}5\).

 Apr 12, 2024

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