Chapter 8.8, Problem 35PPE

To determine

To describe: The error made in factoring the given expression and correct it.

Answer to Problem 35PPE

The error in factoring is that the factorization is incomplete.

The expression $\left(x^3+2x\right)$ should be factorize further.

Explanation of Solution

Given information:

The erroneous factorization of the polynomial is given here:

  $4x^4+12x^3+8x^2+24x=4\left(x^4+3x^3+2x^2+6x\right)$

  $=4\left(x^3\left(x+3\right)+2x\left(x+3\right)\right)$

  $=4\left(\left(x^3+2x\right)\left(x+3\right)\right)$

Formula used:

Factorization is performed using distributive property.

Calculations:

The factorization of polynomial $4x^4+12x^3+8x^2+24x$ is given here:

  $4x^4+12x^3+8x^2+24x=4x\left(x^3+3x^2+2x+6\right)$

  $\Rightarrow 4x^4+12x^3+8x^2+24x=4x\left(x^3\left(x+3\right)+2\left(x+6\right)\right)$

  $\Rightarrow 4x^4+12x^3+8x^2+24x=4x\left(x+3\right)\left(x^3+2\right)$

Thus, the factored polynomial is $4x^4+12x^3+8x^2+24x=4x\left(x+3\right)\left(x^3+2\right)$ not as given $4x^4+12x^3+8x^2+24x=4\left(\left(x^3+2x\right)\left(x+3\right)\right)$ .

The error in factoring is that the factorization is incomplete.

The expression $\left(x^3+2x\right)$ should be factorize further.

Conclusion:

The error in factoring is that the factorization is incomplete.

The expression $\left(x^3+2x\right)$ should be factorize further.