Solve Linear Combinations: -9 - 7x - 15x^2

[_Haemon_]

hi there, my book didn't have an example like this so I am not sure what to do to solve it. Please explain how to do it, thanks.

Express the following as linear combinations of p1 = 2 + x + 4x^2, p2 = 1 - x + 3x^2, and p3 = 3 + 2x + 5x^2

a.) -9 - 7x - 15x^2

 

[Office_Shredder]

You multiply p1 by some constant a, p2 by a constant b, p3 by a constant c. Then comparing the coefficients of the different degrees of x with the polynomial you're trying to get, you have three equations with three unknowns, so you solve for a, b and c

 

[HallsofIvy]

a(2+x+4x^2)+ b(1- x+ 3x^2)+ c(3+ 2x+ 5x^2)= -9- 7x- 15x^2

Solve for a, b, and c so that is true for all x.

There are two ways to do that. One is, since this must be true for all x, to choose three values for x, thus getting 3 equations to solve for a, b, and c. The other is to use the fact that, in order for two polynomials to be equal for all x, "corresponding coefficients" must be equal. Setting corresponding coefficients equal here, again, gives you three equations to solve for a, b, and c.