Find the other polynomial q (x) of the following, given that LCM, GCD and one polynomial p(x) respectively. $(4 x + 5)^{3} (3 x - 7)^{3}, (4 x + 5) (3 x - 7)^{2}, (4 x + 5)^{3} (3 x - 7)^{2}$

(3 x - 7)^{3} (4 x + 5)

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(3 x + 7)^{3} (4 x + 5)

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(3 x - 7)^{3} (4 x - 5)

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None of these

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Solution

The correct option is A $(3 x - 7)^{3} (4 x + 5)$

We know that if $p (x)$ and $q (x)$ are two polynomials, then $p (x) \times q (x) =$ {GCD of $p (x)$ and $q (x)$ } $\times$ {LCM of $p (x)$ and $q (x)$ }.

Now, it is given that one of the polynomial is $p (x) = (4 x + 5)^{3} (3 x - 7)^{2}$ , the LCM is $(4 x + 5)^{3} (3 x - 7)^{3}$ and the GCD is $(4 x + 5) (3 x - 7)^{2}$ , therefore, we have:

$((4 x + 5)^{3} (3 x - 7)^{2}) (q (x)) = (4 x + 5) (3 x - 7)^{2} \times (4 x + 5)^{3} (3 x - 7)^{3}$

$\Rightarrow ((4 x + 5)^{3} (3 x - 7)^{2}) (q (x)) = (4 x + 5)^{4} (3 x - 7)^{5}$

$\Rightarrow q (x) = \frac{(4 x + 5)^{4} (3 x - 7)^{5}}{(4 x + 5)^{3} (3 x - 7)^{2}}$

$\Rightarrow q (x) = (4 x + 5) (3 x - 7)^{3}$

Hence, the other polynomial $q (x)$ is $(3 x - 7)^{3} (4 x + 5)$ .

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Find the zero of the polynomial in each of the following cases:

(i) p(x) = x + 5 (ii) p(x) = x − 5 (iii) p(x) = 2x + 5

(iv) p(x) = 3x − 2 (v) p(x) = 3x (vi) p(x) = ax, a ≠ 0

(vii) p(x) = cx + d, c ≠ 0, c, are real numbers.

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