Find the coefficient of x2 in the following polynomial (2x-5)2×2-3x+1.

# Find the coefficient of ${x}^{2}$ in the following polynomial $\left(2x-5\right)\left(2{x}^{2}-3x+1\right)$.

1. A
16
2. B
4
3. C
-16
4. D
-15

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### Solution:

The given expression is $\left(2x-5\right)\left(2{x}^{2}-3x+1\right)$.
We know that the coefficient of ${x}^{m}$  in $a{x}^{m}+\mathit{bx}+c$ is $a$.
Therefore, write the expression $\left(2x-5\right)\left(2{x}^{2}-3x+1\right)$ in polynomial form,
$\left(2x-5\right)\left(2{x}^{2}-3x+1\right)$
$⇒2x×2{x}^{2}+2x×-3x+2x×1+\left(-5\right)×2{x}^{2}+\left(-5\right)×-2x+\left(-5\right)×1$
$⇒4{x}^{3}-6{x}^{2}+2x-10{x}^{2}+2x-5$
$⇒4{x}^{3}-16{x}^{2}+4x-5$
Therefore, the coefficient of ${x}^{2}$ in is -16.
Hence, option 3 is  correct.

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