Express the polynomial v = x 2 + 4x − 3 in P(x) as a linear combination of the polynomials p1(x) = x 2 − 2x + 5, p2(x) = 2x 2 − 3x and p3(x) = x + 1.

Answer 1

The question assumes that polynomial v is dependent on the three polynomials p₁, p₂, and p₃.

To test validity of the given assumption, we will express v as a linear combination of the p's, as follows:

v = α p₁ + β p₂ + γ p₃ --------------(1)

Where, the coefficients α , β , and γ, are unknow constants, to be determined as follows:

The v and the p's are given in the question as follows:

v = x ² + 4x − 3

and

p₁ (x) = x ² − 2x + 5,

p₂(x) = 2 x ² − 3x.

p₃(x) = x + 1.

Using those given polynomials into equation (1), we get:

x ² + 4x − 3 = α (x ² − 2x + 5) + β ( 2 x ² − 3x ) + γ ( x + 1) -----(2)

Equation (2) will help use determine the three unknowns α , β , and γ, as follows.

We start by equating the coefficients of x ² on both sides. That gives:

1 = α + 2 β + 0 γ ----------------(3 A)

Then we equate the coefficients of x on both sides. That gives:

4 = - 2 α -3 β + γ ----------------------(3 B)

Finally, we equate the constants on both sides. That gives:

− 3 = 5 α + 0 β + γ ---------------(3 C)

The three equations (3 A, B, C) will be solved for the three unknowns, as follows:

First, to eliminate γ, from equations (3 C) and (3 B), subtract equation (3 C) from equation (3 B). That gives

7 = -7 α -3 β --------------( 4 A)

Second, to eliminate α, from equations (3 A) and (4A ),multiply equation (3 A) by 7, then add it to equation (4 A). That gives

14 = 11 β , or β = 14 / 11 -------(4 B)

Having obtained the value of β, we now substitute that in equation (4A) to get the value of α.

1 = - α - 6 /11, or a = - 17/11 --------( 4 C)

The remaining unknown is then determined by equation (3 C) by substituting with the value of a .

γ = (-33 + 85 ) / 11 , or γ - = 52 / 11--------------( 4 D )

That completes our solution, as :

a = - 17/11

β = 14 / 11

γ - = 52 / 11

Substitute by the three values in equation (1) , we get:

v = α p₁ + β p₂ + γ p₃

...= (- 17/11 ) p₁ + (14 / 11 ) p₂ + ( 52 / 11) p₃

Which is the required linear assembly of polynomials of v.