Question

How do you simplify the product $\left( {6x - 5} \right)\left( {3x + 1} \right)$ and write it in standard form?

Answer 1

Hint: The standard form of any quadratic expression is $a{x^2} + bx + c$. This problem deals with expanding algebraic expressions. While expanding an algebraic expression, we combine more than one number or variable by performing the given algebraic operations. The basic steps to simplify an algebraic expression are: remove parentheses by multiplying factors, use exponent rules to remove parentheses in terms with exponents, combine like terms by adding coefficients, and then combine the constants.

Complete step-by-step answer:
Given a product of two linear expression in variable $x$, as shown below:
$ \Rightarrow \left( {6x - 5} \right)\left( {3x + 1} \right)$
Now expanding the given expression by using the distributive property.
$ \Rightarrow \left( {6x - 5} \right)\left( {3x + 1} \right)$
Now simplify in such a way that each term in first polynomial is multiplied with the second polynomial, as shown below:
$ \Rightarrow 6x\left( {3x + 1} \right) - 5\left( {3x + 1} \right)$
Now using the distributive property multiply and simplify and expand each expression, as shown below:
\[ \Rightarrow \left[ {6x\left( {3x} \right) + 6x\left( 1 \right)} \right] - \left[ {5\left( {3x} \right) + 5\left( 1 \right)} \right]\]
\[ \Rightarrow 6x\left( {3x} \right) + 6x\left( 1 \right) - 5\left( {3x} \right) - 5\]
Now simplify the multiplication of each term, of the above expression, as shown below:
\[ \Rightarrow 18{x^2} + 6x - 15x - 5\]
Taking the $x$ term common, in the above expression, and simplifying the above expression further, as shown below:
\[ \Rightarrow 18{x^2} - 9x - 5\]
As we know that the standard form of expression is in the form of $a{x^2} + bx + c$.
So the product of the expression $\left( {6x - 5} \right)\left( {3x + 1} \right)$ in the standard form is equal to \[18{x^2} - 9x - 5\].

$\therefore \left( {6x - 5} \right)\left( {3x + 1} \right) = 18{x^2} - 9x - 5$

Note:
While solving this problem please note that we expanded the given expression with the help of distributive property. This is done by using the distributive property to remove any parentheses or brackets and by combining the like terms and unlike terms. If you see parenthesis with more than one term inside, then distribute first.