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Multiplying monomial with a polynomial:

Let us recall the distributive proportion.

If \(a\) is a constant, \(x\) and \(y\) are variables, then \(a(x + y) = ax + ay\).

Example:

**1.**Suppose there will be \(x\) number of bags and a bag contains \(3\) cupcakes of '\(p\)' packs, \(7\) chocolates of '\(q\)' packs and \(5\) cookies of '\(r\)' packs. The total number of items can be identified by adding the number of items in the bag and product with the number of bags.

This can be written as $x(3p+7q+5r)$.

Applying the distributive property,

\(= 3px + 7qx + 5rx\).

**2.**Find the product of $3{p}^{3}q$ and $\left(2{\mathit{pq}}^{3}-5{p}^{2}+3{q}^{4}\right)$.

$=\phantom{\rule{0.147em}{0ex}}3{p}^{3}q\times \left(2{\mathit{pq}}^{3}-5{p}^{2}+3{q}^{4}\right)$

Applying the distributive property,

\(=\) \((3×2)\) \(p^{3+1}q^{1+3}\) \(+\) \((3×-5)\)\(p^{3+2}q\) \(+\) \((3×3)p^3q^{1+4}\)

\(=\) \(6p^4q^4-15p^5q+9p^3q^5\).