Find the product of $(4 x^{2} - 9)$ and $(2 x^{2} - 3 x + 1)$ and divide the product by $(4 x^{3} - 7 x + 3) .$ Is $(4 x^{3} - 7 x + 3)$ a factor of the product?

True

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False

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Solution

The correct option is A True
First, let us find the product.
$(4 x^{2} - 9) (2 x^{2} - 3 x + 1)$
$= 4 x^{2} (2 x^{2} - 3 x + 1) - 9 (2 x^{2} - 3 x + 1)$
$= 8 x^{4} - 12 x^{3} - 14 x^{2} + 27 x - 9$

Now, we need to divide the product, i.e; $= 8 x^{4} - 12 x^{3} - 14 x^{2} + 27 x - 9$ by $(4 x^{3} - 7 x + 3)$
This means we have to divide $(4 x^{2} - 9) (2 x^{2} - 3 x + 1)$ by $(4 x^{3} - 7 x + 3)$
Now, $(4 x^{2} - 9) (2 x^{2} - 3 x + 1)$
$= (2 x - 3) (2 x + 3) (2 x^{2} - 3 x + 1)$
$= (2 x - 3) (4 x^{3} - 7 x + 3)$

So, $(4 x^{3} - 7 x + 3)$ is a factor of $(4 x^{2} - 9) (2 x^{2} - 3 x + 1)$

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