Answer to Question #223044 in Linear Algebra for Susan

Question

Write the polynomial x^2+4x-3 as a linear combination of 2x^2-3x and x^2-2x+5 and x+3

Answer 1

Let

$x^2+4x-3=a(2x^2-3x)+b(x^2-2x+5)+c(x+3)$

$x^2+4x-3=(2a+b)x^2+(-3a-2b+c)x+(5b+3c)$

$x^2:2a+b=1$

$x^1:-3a-2b+c=4$

$x^0:5b+3c=-3$

$a=\dfrac{1}{2}-\dfrac{1}{2}b$

$c=-1-\dfrac{5}{3}b$

$-\dfrac{3}{2}+\dfrac{3}{2}b-2b-1-\dfrac{5}{3}b=4$

$-\dfrac{13}{6}b=\dfrac{13}{2}$

$b=-3$

$a=2$

$c=4$

$x^2+4x-3=2(2x^2-3x)-3(x^2-2x+5)+4(x+3)$