Write the polynomial q(x) =2x^-10x+5 as a linear combination of the polynomials p1(x) =x^2+2x-3,p2(x) =2x^2+2x+3,p3(x) =3x^2-2x+1

Question

Write the polynomial q(x) =2x^-10x+5 as a linear combination of the polynomials p1(x) =x^2+2x-3,p2(x) =2x^2+2x+3,p3(x) =3x^2-2x+1

Answer 1

Find numbers a, b, and c so that a(x²+2x-3) + b(2x²+2x+3) + c(3x²-2x+1) = 2x²-10x+5

Then (a+2b+3c)x² + (2a+2b-2c)x + (-3a+3b+c) = 2x² -10x + 5

Equating coefficients of like powers, we have the system of equations:

a + 2b + 3c = 2

2a + 2b - 2c = -10

-3a + 3b + c = 5

Add -2 times the first equation to the second: -2b - 8c = -14

Add 3 times the first equation to the third: 9b +10c = 11

Divide -2b-8c = -14 by -2: b + 4c = 7

9b + 10c = 11

Add -9 times the first equation to the second: -26c = -52

So, c = 2

Backsubstitute to obtain b = -1 and a = -2.

So, -2p₁(x) - 1p₂(x) + 2p₃(x) = q(x)