Find the coefficient of $x$ and $x^{2}$ $(x - 5) (x - 6) (x - 1)$

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Solution

The given binomials are $(x - 5)$ , $(x - 6)$ and $(x - 1)$ .

Firstly multiply the first and second term as follows:

$[(x - 5) (x - 6)] (x - 1) = [(x \times x) + (x \times - 6) + (- 5 \times x) + (- 5 \times - 6)] (x - 1) = (x^{2} - 6 x - 5 x + 30) (x - 1)$
$= (x^{2} - 11 x + 30) (x - 1)$

Now multiply the resulting expression:

$(x^{2} - 11 x + 30) (x - 1) = (x^{2} \times x) + (x^{2} \times - 1) + (- 11 x \times x) + (- 11 x \times - 1) + (30 \times x) + (30 \times - 1)$
$= x^{3} - x^{2} - 11 x^{2} + 11 x + 30 x - 30 = x^{3} - 12 x^{2} + 41 x - 30$

Therefore, $(x - 5) (x - 6) (x - 1) = x^{3} - 12 x^{2} + 41 x - 30$ .

Hence, the coefficient of $x^{2}$ is $- 12$ and coefficient of $x$ is $41$ .

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