Question

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.

Answer 1

-23/2 or -11.5

Explanation:

We find the expansion of

(x^3 - 4x^2 + 2x - 5)(x^2 + tx - 7)

by multiplying each term in x^3 - 4x^2 + 2x - 5 by each term in x^2 + tx - 7, and then summing the resulting products. We only care about the x^2 term, so we focus solely on the products that produce an x^2 term. Thus, the x^2 term in the product is equal to

(-4x^2)(-7) + (2x)(tx) + (-5)(x^2) = 28x^2 + 2tx^2 - 5x^2 = (2t + 23)x^2.

Hence, if the product has no x^2 term, then the coefficient of x^2, namely 2t + 23, must be equal to 0. This occurs when t = -23/5.

Answer 2

The expansion of the product of the given polynomials can be determined through the distributive property of multiplication.

(x³ - 4x² + 2x - 5)(x² + tx - 7)

This goes with the first term of the first expression first up until every term of the first expression is multiplied to the terms of the second expression.

x⁵ + tx⁴ - 7x³ - 4x⁴ - 4tx³ + 28x² + 2x³ + 2tx² - 14x - 5x² - 5tx + 35

The terms with x² are 28x², 2tx², and -5x². The sum of the numerical coefficients should be zero.
28 + 2t - 5 = 0

The value of t from the equation is -11.5.