Question

Multiply: \[\left( ax+b \right)\] by \[\left( cx+d \right)\]

Answer 1

Hint: In this type of question we have to use the concept of multiplication of algebraic expressions. In the given question when we perform multiplication we have to multiply the second expression by the first term of first expression and then multiply the second expression by the second term of first expression. We can perform the multiplication in reverse order also, that is we can multiply the first expression by the first term of second expression and then multiply the first expression by the second term of second expression. In both ways we get the same result.

Complete step-by-step answer:
Now we have to multiply \[\left( ax+b \right)\] by \[\left( cx+d \right)\] which we can represent as
\[\Rightarrow \left( ax+b \right)\times \left( cx+d \right)\]
Let us first multiply the second expression by the first term of first expression and then multiply the second expression by the second term of first expression.
Hence in this case we have to multiply \[\left( cx+d \right)\] by \[ax\] first and then we multiply \[\left( cx+d \right)\] by \[b\] so that we get,
\[\Rightarrow \left( ax+b \right)\times \left( cx+d \right)=ax\times \left( cx+d \right)+b\times \left( cx+d \right)\]
On simplifying we can write,
\[\Rightarrow \left( ax+b \right)\times \left( cx+d \right)=\left( ax\times cx \right)+\left( ax\times d \right)+\left( b\times cx \right)+\left( b\times d \right)\]
Now, as \[x\times x\] results into \[{{x}^{2}}\] we can write
\[\Rightarrow \left( ax+b \right)\times \left( cx+d \right)=ac{{x}^{2}}+adx+bcx+bd\]
By using the rules of addition we can simplify the above expression as
\[\Rightarrow \left( ax+b \right)\times \left( cx+d \right)=ac{{x}^{2}}+\left( ad+bc \right)x+bd\]
Hence, by performing multiplication of \[\left( ax+b \right)\] by \[\left( cx+d \right)\]we get the expression \[ac{{x}^{2}}+\left( ad+bc \right)x+bd\].

Note: In this type of question students have to note that though the position of expression gets interchanged we will get the same result. Also students have to note that we can combine the constants for the same variable only.