Find the coefficients of $x^{2}$ and $x$ in $(2 x + 1) (2 x - 2) (2 x - 5)$

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Solution

The given binomials are $(2 x + 1)$ , $(2 x - 2)$ and $(2 x - 5)$ .

Firstly multiply the first and second term as follows:

$(2 x + 1) (2 x - 2) (2 x - 5) = [(2 x \times 2 x) + (2 x \times - 2) + (1 \times 2 x) + (1 \times - 2)] (2 x - 5) = (4 x^{2} - 4 x + 2 x - 2) (2 x - 5)$
$= (4 x^{2} - 2 x - 2) (2 x - 5)$

Now multiply the resulting expression:

$(4 x^{2} - 2 x - 2) (2 x - 5) = (4 x^{2} \times 2 x) + (4 x^{2} \times - 5) + (- 2 x \times 2 x) + (- 2 x \times - 5) + (- 2 \times 2 x) + (- 2 \times - 5)$
$= 8 x^{3} - 20 x^{2} - 4 x^{2} + 10 x - 4 x + 10 = 8 x^{3} - 24 x^{2} + 6 x + 10$

Therefore, $(2 x + 1) (2 x - 2) (2 x - 5) = 8 x^{3} - 24 x^{2} + 6 x + 10$ .

Hence, the coefficient of $x^{2}$ is $- 24$ and coefficient of $x$ is $6$ .

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