Q. 366 - Intermediate Algebra

Question

Factor Trinomials of the Form $a{x}^2+bx+c$ Using Trial and Error
In the following exercises, factor completely using trial and error.

$5x^4+10x^3-75x^2$

Step-by-Step Solution

Answer

By factoring the polynomial $5x^4+10x^3-75x^2$ completely by trial and error method, the factors are $5x^2(x-3)(x+5)$

Step 1. Given

The polynomial is $5x^4+10x^3-75x^2$

To factor the polynomial by trial and error method.

Step 2. Factor any GCF

Factor the GCF.

$5x^4+10x^3-75x^2=5x^2(x^2+2x-15)$

Step 3. Find all the pairs of factors of first term

The factors of the first term is $1,1$

Since there is only one term, we can put the in parentheses.
$x^2+2x-15=(x )(x )$

Step 4. Find all the factor pairs of third term

Consider the sign.

Since the last term is $-15$, the factors have opposite signs.

Since the sign of the second term is positive, the greater factor have positive sign. 

The factors of $-15$ are

• $-1,15$
  
• $-3,5$
  

Step 5. Test all combination of factors

Test all combination of factors until the correct product is found. 

Possible factors	Product
$(x-1)(x+15)$	$x^2+14x-15$
$(x-3)(x+5)$	$x^2+2x-15$

The factors of $x^2+2x-15$ is $(x-3)(x+5)$

Step 6. Check the answer

Check by multiplying the factor. 

$5x^2(x-3)(x+5)=5x^2(x^2+2x-15)$

                           $=5x^4+10x^3-75x^2$