Question

How do you simplify $\left( { - 12{x^3} + 5x - 23} \right) - \left( {4{x^4} + 31 - 9{x^3}} \right)$

Answer 1

Hint: Given the polynomial expression. We have to express the polynomial in simplified form. First, rewrite the expression in descending order of degree of expression. Then, we will remove the parentheses by applying the distributive property. Then, combine the like terms and write the expression in simplified form.

Complete step by step solution:
We are given the polynomial expression, $\left( { - 12{x^3} + 5x - 23} \right) - \left( {4{x^4} + 31 - 9{x^3}} \right)$.
First, we will rewrite the expression as descending order of degree.
$\left( { - 12{x^3} + 5x - 23} \right) - \left( {4{x^4} - 9{x^3} + 31} \right)$
Then, subtract the polynomial from the other by reversing the signs of the terms in the polynomial that will be subtracted.
$ \Rightarrow - 12{x^3} + 5x - 23 - 4{x^4} + 9{x^3} - 31$
Now, rearrange the like terms in the expression.
$ \Rightarrow - 12{x^3} + 9{x^3} + 5x - 4{x^4} - 23 - 31$
Now, combine the coefficients of like terms by performing required arithmetic operation, we get:
$ \Rightarrow - 4{x^4} - 3{x^3} + 5x - 54$

Additional Information:
The polynomial is defined as the expression with more than one nonzero term. In the polynomial, the degree is defined as the highest exponent of the variable in the expression. The polynomial contains variable, constant and exponent. The polynomials are of three types which are classified according to the degree of the expression such as linear, cubic and quadratic. The polynomials are involved in basic arithmetic operations such as addition, subtraction, multiplication or division. The subtraction of two polynomials is the arithmetic operation in which the signs of the terms of the second polynomial is reversed and then the coefficients of the similar degree are added or subtracted with each other.

Note:
In such types of questions, students must be familiar with all the arithmetic operations and corresponding definitions. Please note that the subtraction or addition of two polynomials will provide the same degree of polynomial.