Question

How do we write $ 3{x^2}(2{x^3} - 4{x^2}) $ in standard form?

Answer 1

Hint: The given equation is $ 3{x^2}(2{x^3} - 4{x^2}) $ . The formula for expanding
$ a(b + c) = ab + ac $ . After that we apply the expansion of \[a(b + c)\] .If we multiply the two variables, then add the powers. .After that we simplify the equation.
Finally we get the standard form.

Complete step by step answer:
The given equation is $ 3{x^2}(2{x^3} - 4{x^2}) $
The formula for expanding $ a(b + c) $ is $ ab + ac $ . In this expand $ a = 3{x^2};b = 2{x^3}; $ and $ c = - 4{x^2} $
Apply in the expansion of $ a(b + c) $ , hence we get
$ 3{x^2}(2{x^3}) + 3{x^2}( - 4{x^2}) $
To multiply $ a{x^b} $ by $ c{x^d} $ , we do $ (a \cdot c){x^{b + d}} $
Putting our values in, we get $ (3 \cdot 2){x^{2 + 3}} $
Now we multiply $ 3 $ by $ 2 $ and we add the power term, hence we get
$ (3 \cdot 2){x^{2 + 3}} = 6{x^5} $
In the second term $ 3{x^2}( - 4{x^2}) $ , hence we get
$ (3.( - 4)){x^{2 + 2}} $
Now we multiply $ 3 $ by $ - 4 $ and we add the power term, hence we get
$ (3.( - 4)){x^{2 + 2}} = - 12{x^4} $
By add both values together, hence we get
$ 6{x^5} - 12{x^4} $

we write the final standard form $ 6{x^5} - 12{x^4} $

Note: Solving higher order polynomial equations is an essential skill for anybody studying science and mathematics. However, understanding how to solve these kinds of equations is quite challenging. The standard form for writing down a polynomial is to put the terms with the highest degree first (like the $ 2 $ in $ {x^2} $ if there is one).
In geometry a degree $ {(^ \circ }) $ is a way of measuring angles.
But here we look at what a degree means in algebra. In algebra degree is sometimes called order
The degree (for a polynomial with one variable, like $ X $ )is the largest exponent of that variable
We know the degree we can also give it a name.

Degree	Name	Example
$ 0 $	Constant	$ 7 $
$ 1 $	Linear	$ x + 3 $
$ 2 $	Quadratic	$ {x^2} - x + 2 $
$ 3 $	Cubic	$ {x^3} - {x^2} + 3 $
$ 4 $	Quartic	$ 6{x^4} - {x^3} + x - 2 $
$ 5 $	Quintic	$ {x^5} - 3{x^3} + {x^2} + 8 $