Let f(x) = 2x^2 + 5x + 1. If we write f(x) as | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given, $f(x) = 2x^2 + 5x + 1$. Also, $f(x) = a(x+1)(x-2) + b(x-2)(x-1) + c(x-1)(x+1)$. Expanding the right side: $f(x) = a(x^2 - x - 2) + b(x^2 -

Vedclass · Accepted Answer

exactly one choice for each of $a, b, c$

Vedclass · Answer

(C) Given, $f(x) = 2x^2 + 5x + 1$. Also, $f(x) = a(x+1)(x-2) + b(x-2)(x-1) + c(x-1)(x+1)$. Expanding the right side: $f(x) = a(x^2 - x - 2) + b(x^2 - 3x + 2) + c(x^2 - 1)$ $f(x) = (a+b+c)x^2 + (-a-3b)x + (-2a+2b-c)$. On equating the coefficients of $x^2$, $x$, and the constant term, we get a system of linear equations: $1) \; a + b + c = 2$ $2) \; -a - 3b = 5$ $3) \; -2a + 2b - c = 1$ Since we have $3$ linear equations in $3$ variables $(a, b, c)$ and the determinant of the coefficient matrix is non-zero, there exists a unique solution for $a, b, c$. Solving these, we get $a = -\frac{35}{4}$, $b = \frac{5}{4}$, and $c = \frac{38}{4}$. Thus, there is exactly one choice for each of $a, b, c$.

Let $f(x) = 2x^2 + 5x + 1$. If we write $f(x)$ as $f(x) = a(x+1)(x-2) + b(x-2)(x-1) + c(x-1)(x+1)$ for real numbers $a, b, c$, then:

Similar Questions

The equations $x+y+z=3$, $x+2y+2z=6$ and $x+ay+3z=b$ have

If $x=a, y=b, z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$, $x-y+z=2$, and $x+2y+2z=1$, then $ab+bc+ca=$

Let $A=\begin{bmatrix} 0 \\ -6 \\ 8 \end{bmatrix}$,$B=\begin{bmatrix} 3 & 5 & -7 \\ 0 & -1 & 8 \\ 6 & -1 & 0 \end{bmatrix}$ and $X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $D=[\alpha, \beta, \gamma]^{T}$ is the solution of $X^{T} B^{T}=A^{T}$,then $D^{T} A=$

For the system of linear equations $a x+y+z=1$,$x+a y+z=1$,$x+y+a z=\beta$,which one of the following statements is $NOT$ correct?

If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions,then $\alpha + \beta - \alpha \beta$ is equal to .... .

Vedclass Test Series

Exam Paper Generator

Online Exam Module