Chapter 8.7, Problem 4BGP

To determine

To factor: The given polynomial.

Answer to Problem 4BGP

The factored polynomial is $12x^2+5x-25=\left(3x+5\right)\left(4x-5\right)$ .

Explanation of Solution

Given information:

The given polynomial is $12x^2+5x-25$ .

Concept used:

For polynomial $a{x}^2+bx-c$ , find two factors of $a\times c$ such that their difference is b.

Calculations:

The given polynomial E (say) is $E=12x^2+5x-25$ that can be written as

We find two factors of $12\times 25$ such that their difference is 5 as follows:

  $12\times 25=2\times 2\times 3\times 5\times 5$

  $\Rightarrow 12\times 25=\left(3\times 5\right)\left(2\times 2\times 5\right)$

  $\Rightarrow 12\times 25=\left(15\right)\left(20\right)$

Such that $20-15=5$ .

Thus, the polynomial $E$ can be written as $E=12x^2+\left(20-15\right)x-25$

  $\Rightarrow E=12x^2+20x-15x-25$

  $\Rightarrow E=4x\left(3x+5\right)-5\left(3x-5\right)$

  $\Rightarrow E=\left(3x+5\right)\left(4x-5\right)$

Thus, the factored polynomial is $E=12x^2+5x-25=\left(3x+5\right)\left(4x-5\right)$ .

Conclusion:

The factored polynomial is $12x^2+5x-25=\left(3x+5\right)\left(4x-5\right)$ .