Q 110. - Intermediate Algebra

Question

Factor Trinomials of the Form $a{x}^2+bx+c$ Using Trial and Error.

$-5y^3-30y^2+35y$

Step-by-Step Solution

Answer

The solution is $-5y(y-1)(y+7)$.

Step 1. Given information

Consider the trinomial.

$-5y^3-30y^2+35y$

Step 2. Write the trinomial in descending order and then find the greatest common factor.

The trinomial $-5y^3-30y^2+35y$ is given in descending order.

There is a greatest common factor in the given trinomial.

Factor the greatest common factor.

$-5y^3-30y^2+35y=-5y(y^2+6y-7)$

Step 3. Find the factors of the first term and the last term of the trinomial y 2 + 6 y - 7 .

The first term of the trinomial $y^2+6y-7$ is $y^2$.

So, the factors of $y^2$ are as follows:

$y^2=y·y$

The last term of the trinomial $y^2+6y-7$ is $-7$.

Since the last term of the given polynomial is negative, the factors of the last term will have opposite signs. 

The factors of $-7$ are as follows:

$\begin{array}{rcl}-7& =& 1·(-7)\\ -7& =& (-1)·7\end{array}$

Step 4. Make a table for all the combination of factors of y 2 + 6 y - 7 .

Possible factors	Product
$(y+1)(y-7)$	$y^2-6y-7$
$(y-1)(y+7)$	$y^2+6y-7$

From the table, conclude that the combination  $(y-1)(y+7)$ is correct. 

Step 5. Check by multiplying all the factors - 5 y ( y - 1 ) ( y + 7 ) .

$\begin{array}{rcl}-5y(y-1)(y+7)& =& -5y(y^2+7y-y-7)\\ & =& -5y(y^2+6y-7)\\ & =& -5y^3-30y^2+35y\end{array}$

Hence, the factor is $-5y(y-1)(y+7)$.