At least one of the factors of the polynomial equation $P (x) = 3 x^{2} + 10 x + 7$ divides which of the following polynomial equations?

x³ + 2x² + 3x + 2

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x⁴ + 3x³ + x² + x +2

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7x⁵ + 4x⁴ + 5x³ + 8x² + x + 1

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All of these

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Solution

The correct option is D
All of these

Let the given polynomial be $P (x) = 3 x^{2} + 10 x + 7 = 0$

$3 x^{2} + 10 x + 7$ can be written as $3 x^{2} + 3 x + 7 x + 7 = 0$

$3 x^{2} + 3 x + 7 x + 7 = 3 x (x + 1) + 7 (x + 1) = (x + 1) (3 x + 7)$

So, the factors would be $(x + 1)$ and $(3 x + 7)$ .

Check whether $(x + 1)$ , $(3 x + 7)$ can be factors of the polynomials given in the options by using factor’s theorem.

You can see that the value of all polynomials given in the options is zero at x=-1.

So by factor’s theorem, we can say that $(x + 1)$ is a factor of all the polynomials in the options. Thus, D is the correct option.

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