Chapter 8.2, Problem 50MR

To determine

To simplify: The given sum $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)$ .

Answer to Problem 50MR

The simplified form of the expression $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)$ is $7x^4+3x^2-1$ .

Explanation of Solution

Given information:

The given sum is $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)$ .

Concept used:

The sum of two or more polynomials is equal the sum of coefficients of like powers of variables.

Example: $\left(a{x}^2+b{x}^3+c\right)+\left(dx+e{x}^2\right)$ = $b{x}^3+a{x}^2+e{x}^2+dx+c=b{x}^3+\left(a+e\right)x^2+dx+c$

Calculations:

The given sum $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)$ may be written as

  $4x^4+3x^4-3x^2+6x^2-1$

Collecting the terms of like powers of x

  $\Rightarrow \left(4+3\right)x^4-\left(3-6\right)x^2-1$

Adding coefficients of like powers of x

  $\Rightarrow 7x^4-\left(-3\right)x^2-1$

  $\Rightarrow 7x^4+3x^2-1$

Simplified result

Thus, we have $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)=7x^4+3x^2-1$

Thus, the simplified form of the expression $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)$ is $7x^4+3x^2-1$ .

Conclusion:

The simplified form of the expression $\left(4x^4-3x^2-1\right)+\left(3x^4+6x^2\right)$ is $7x^4+3x^2-1$ .