Multiplying a Monomial by a Polynomial

Carry out the multiplication of the expression in the following pair:

$\mathrm{a}+\mathrm{b}, 7{\mathrm{a}}^2\mathrm{b}$

Carry out the multiplication of the expression in the following pairs:
$\mathrm{ab}, \mathrm{a}-\mathrm{b}$

Carry out the multiplication of the expression in each of the following pairs:
$4\mathrm{p}, \mathrm{q}+\mathrm{r}$

On subtracting $3\mathrm{a}(\mathrm{a}+\mathrm{b}+\mathrm{c})–2\mathrm{b}(\mathrm{a}–\mathrm{b}+\mathrm{c})$ from $4\mathrm{c}(–\mathrm{a}+\mathrm{b}+\mathrm{c})$ if we get $-3a^2-2b^2+4c^2-ab+kbc-7ac$. 

Then find the value of $k$.

On subtracting $3\mathrm{l}(\mathrm{l}–4\mathrm{m}+5\mathrm{n})$ from $4\mathrm{l}(10\mathrm{n}–3\mathrm{m}+2\mathrm{l})$ if we get $k{l}^2+25\ln$. 

Then find the value of $k$.

On adding $2\mathrm{x}(\mathrm{z}–\mathrm{x}–\mathrm{y}) \mathrm{and} 2\mathrm{y}(\mathrm{z}–\mathrm{y}–\mathrm{x})$ if we get $-2x^2-2y^2-4xy+kz+2zx$. 

Then find the value of $k$.

On adding $\mathrm{p}(\mathrm{p}-\mathrm{q}), \mathrm{q}(\mathrm{q}–\mathrm{r})$and$\mathrm{r}(\mathrm{r}–\mathrm{p})$if we get $p^2+q^2+r^2-pq-qr+k$.

Then find the value of $k$.

Simplify  $\mathrm{a}\left({\mathrm{a}}^2+\mathrm{a}+1\right)+5$ and find the value for  $\mathrm{a}=-1$.
 

Simplify $\mathrm{a}\left({\mathrm{a}}^2+\mathrm{a}+1\right)+5$ and find the value for $\mathrm{a}=1$ 

Simplify $\mathrm{a}\left({\mathrm{a}}^2+\mathrm{a}+1\right)+5$ and find the value for $\mathrm{a}=0$.
 

Simplify  $3\mathrm{x} \left(4\mathrm{x}-5\right)+3$ and find its value for  $\mathrm{x}=\frac{1}{2}$ 
 

If the product of $x\times x^2\times x^3\times x^4$  is of the form $x^m$, then find the value of $m$.

If the product $\left(-\frac{10}{3}{\mathrm{pq}}^3\right)\times \left(\frac{6}{5}{\mathrm{p}}^3\mathrm{q}\right)=-4p^k{q}^4$.

Then find the value of $k$.

If the product $\left(\frac{2}{3}\mathrm{xy}\right)\times \left(-\frac{9}{10}{\mathrm{x}}^2{\mathrm{y}}^2\right)=-\frac{3}{5}{x}^k{y}^3$

If the product $\left({\mathrm{a}}^2\right)\times \left(2{\mathrm{a}}^{22}\right)\times \left(4{\mathrm{a}}^{26}\right)=k{a}^m$, then find the value of $k$.

The product of $a+b+c$ and $abc$ is $a^{2}bc+a{b}^{2}c+ab{c}^k$. Find $k$.

Multiplying $4p^2{q}^2, p^2-q^2$, we get, $4p^k{q}^2-4p^2{q}^4$. Find $k$.

Complete the table.

Find the value of $k$.