Question

How do you multiply \[3x\left( {5{x^2} - x + 4} \right)\]?

Answer 1

Hint: We will use the concepts of polynomials and algebraic expressions to solve this problem. We will know about polynomials in detail while solving this problem. We will know about like terms and unlike terms and also about multiplication and divisions of polynomials using some standard formulas.

Complete answer:
In algebra, a variable is a term whose value will be constantly changing according to situations and conditions. It is generally represented as \[x,y,z,a,b,c,.....\]
So, an expression containing variables and powers of variables is called a ‘polynomial’.
For example, take \[{x^3} + 4{y^6} - \dfrac{1}{7}z\]. This is a polynomial.
Like terms can be added or subtracted and can be simplified.
For example, \[4x + 7x = 11x\].
Adding or subtracting like terms will also give another like term.
Like terms are the terms having the same variable. Take example \[7xy\] and \[\dfrac{{ - 9}}{2}xy\]. These two terms have the same variables. So, these two terms are like terms.
But, take \[6xy\] and \[ - 5{x^2}y\]. These two terms have the same variables, but with different powers. So, these two terms are not like terms. These two are unlike terms.
But, to multiply variables, we do not need to bother about like terms or unlike terms.
Some examples for multiplications are: -
\[4a \times 7b = 28ab\]
\[\dfrac{b}{2} \times {b^2} = \dfrac{{{b^3}}}{2}\]
Now, in the question, it is given as \[3x\left( {5{x^2} - x + 4} \right)\]
So, here we will apply distributive law, which states that, \[a \times (b + c) = a \times b + a \times c\]
So, we can write it as
\[ \Rightarrow 3x\left( {5{x^2} - x + 4} \right) = 3x\left( {5{x^2}} \right) - 3x\left( x \right) + 3x\left( 4 \right)\]
\[ \Rightarrow 3x\left( {5{x^2} - x + 4} \right) = 15{x^3} - 3{x^2} + 12x\]
So, like this we can multiply the terms.

Note:
If power in a polynomial is fractional, then it is not a polynomial.
We can use some algebraic properties or formulas like \[{a^m} \times {a^n} = {a^{m + n}}\] (product of exponents with same base) and \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] (division of exponents with same base)
\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
So, we can use these formulas for simplifying polynomial expressions.