Chapter 8.8, Problem 38P

Error Analysis Describe and correct the error made in factoring completely.

To determine

To describe and correct: The error made in factoring a polynomial.

Answer to Problem 38P

The error in factoring the polynomial is “one factor is not simplified completely, and the completely factored polynomial is $16x^4+24x^3+64x^2+96x=8x\left(2x^2+8\right)\left(2x+3\right)$ .

Explanation of Solution

Given information:

The given factored polynomial is $16x^4+24x^3+64x^2+96x=4x\left(2x^2+8\right)\left(2x+3\right)$ .

Concept used:

The multiplication of factors gives the original polynomial.

Calculations:

The given factored polynomial is

  $16x^4+24x^3+64x^2+96x=4x\left(2x^2+8\right)\left(2x+3\right)$.............(1)

The multiplication of factors on Right Hand Side of gives

  $4x\left(2x^2+8\right)\left(2x+3\right)=4x\left(4x^3+6x^2+16x+24\right)$

  $\Rightarrow 4x\left(2x^2+8\right)\left(2x+3\right)=16x^4+24x^3+64x^2+96x$

Thus, the multiplication of factors trace the original polynomial. Therefore, check whether each factor is simplified completely or not.

It is observed that the factor $\left(2x^2+8\right)$ is not simplified completely. That is the error as its further simplification is $\left(2x^2+8\right)=2\left(x^2+4\right)$ .

Hence, the completely factored polynomial is $16x^4+24x^3+64x^2+96x=4x\cdot 2\left(2x^2+8\right)\left(2x+3\right)$

  $\Rightarrow 16x^4+24x^3+64x^2+96x=8x\left(2x^2+8\right)\left(2x+3\right)$ .

Conclusion:

The error in factoring the polynomial is “one factor is not simplified completely, and the completely factored polynomial is $16x^4+24x^3+64x^2+96x=8x\left(2x^2+8\right)\left(2x+3\right)$ .