Question

How do you simplify $\left( 3x+4 \right)\left( 2x+3 \right)$) ? \[\]

Answer 1

Hint: We recall the distributive property of multiplication over addition. We multiply the given polynomials using distributive property $\left( a+b \right)\left( c+d \right)=a\left( c+d \right)+b\left( c+d \right)$. We again use distributive property and then simplify. \[\]

Complete step by step answer:
We know there are four properties for the arithmetic operation called multiplication called closure, commutative, associative and distributive. The distributive property of multiplication requires one more operation either addition or subtraction.
We know that the distributive property of multiplication over addition states that the product of a number say $a$ with sum of two numbers say $b,c$ is equal to the sum of products of the number $a$ multiplied separately with $b$ and $c$. It means
\[\begin{align}
  & a\times \left( b+c \right)=a\times b+a\times c \\
 & \Rightarrow a\left( b+c \right)=ab+ac \\
\end{align}\]
Here we give a parenthesis around $b+c$ to prioritize addition over multiplication so that we can find the sum first. It is called distributive property because the number $a$ outside the bracket is being distributed by each number inside the bracket which are$b,c$. If we have $\left( a+b \right)\left( c+d \right)$ we can distribute $\left( c+d \right)$ with $a,b$ to have
\[\left( a+b \right)\left( c+d \right)=a\left( b+c \right)+b\left( c+d \right)\]
We are asked in the question to simplify
\[\left( 3x+4 \right)\left( 2x+3 \right)\]
We see that the above polynomials are in the form $\left( a+b \right)\left( c+d \right)$. We use distributive property for $a=3x,b=4,c=2x,d=3$ to multiply as -
\[\begin{align}
  & \Rightarrow \left( 3x+4 \right)\left( 2x+3 \right) \\
 & \Rightarrow 3x\left( 2x+3 \right)+4\left( 2x+3 \right) \\
\end{align}\]
We again use distributive property in the form $a\left( b+c \right)=ab+ac$ in the above step for $a=3x,b=2x,c=3$ in the first term and then for $a=4,b=2x,c=3$ in the second term to multiply as ;
\[\begin{align}
  & \Rightarrow 3x\times 2x+3x\times 3+4\times 2x+4\times 3 \\
 & \Rightarrow 6{{x}^{2}}+9x+8x+12 \\
\end{align}\]
 We add the like terms to have the simplified expression-
\[\Rightarrow 6{{x}^{2}}+17x+12\]

Note:
We note that the polynomial with degree 1 is called linear polynomial and the polynomial with degree 2 is called quadratic polynomial. The polynomial we multiplied here $3x+4,2x+3$are linear and the product is quadratic. We can alternatively distribute $a+b$ over $c,d$ as $\left( a+b \right)\left( c+d \right)=\left( a+b \right)c+\left( a+b \right)d$.