Q. 364 - 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.

$x^3+5x^2-24x$

Step-by-Step Solution

Answer

By factoring the polynomial $x^3+5x^2-24x$ completely by trial and error method, the factors are $x(x-3)(x+8)$

Step 1. Given

The polynomial is $x^3+5x^2-24x$

To factor the polynomial by trial and error method.

Step 2. Factor any GCF

Factor the GCF

$x^3+5x^2-24x=x(x^2+5x-24)$

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.

$x^2+5x-24=(x )(x )$

Step 4. Find all the factor pairs of third term

Consider the sign.

Since the last term is $-24$, the factors have opposite signs.

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

The factors of $24$ are

• $-1,24$
  
• $-2,12$
  
• $-3,8$
  
• $-4,6$
  

Step 5. Test all combination of factors

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

Possible factors	Product
$(x-1)(x+24)$	$x^2+23x-24$
$(x-3)(x+8)$	$x^2+5x-24$
$(x-2)(x+12)$	$x^2+10x-24$
$(x-4)(x+6)$	$x^2+2x-24$

So the factors are $x^2+5x-24=(x-3)(x+8)$

Step 6. Check the answer

Check by multiplying the factor.

$x(x-3)(x+8)=x(x^2+5x-24)$

                       $=x^3+5x^2+24x$