How do you factor #3w ^ { 6} - 26w ^ { 5} + 16w ^ { 4}#?

2 Answers
Meave60
Nov 17, 2017

#3w^6-26w^5+16w^4=color(blue)(w^4(3w-2)(w-8)#

Explanation:

Factor:

#3w^6-26w^5+16w^4#

Factor out the common #w^4#.

#w^4(3w^2-26w+16)#

The rest of the polynomial will be factored by splitting the middle method.

Multiply the coefficient of the first term of #3w^2-26w+16# by the constant.

#3xx16=48#

Find two numbers that when added equal #-26# and multiply to #48#. The numbers #-2# and #-24# meet the requirements.

Split #-26# as the sum of #-2w# and #-24w#.

#3w^2-2w-24w+16#

Factor out common terms in first two terms and the last two terms.

#w(3w-2)-8(3w-2)#

Factor out the common term #(3w-2)#

#(3w-2)(w-8)#

Return the factor #w^4#.

#w^4(3w-2)(w-8)#

Shantelle
Nov 17, 2017

#w^4(3w-2)(w-8)#

Explanation:

#3w^6 - 26w^5 + 16w^4#
First, we factor out #w^4# since that is the only thing that they all have, so it looks like this:
#w^4(3w^2 - 26w + 16)#

Let's just look at #3w^2 - 26w + 16# now because that is the part of the expression we are trying to factor now.

This is also known as Standard Form for a quadratic, #ax^2 + bx + c#.

In order to attempt to factor this using the grouping method, we first need to find a number that will multiply to #a*c# and add up to #b#.

When we multiply #3# and #16#, we get #48#. So we need to find a number that multiplies to #48# and adds up to #-26#. #-24# and #-2# work because #-24 * -2 = 48# and #-24 - 2 = -26#. Using these two numbers #-24# and #-2#, we put them into the equation for #w#.

So now our expression looks like this:
#3w^2 - 24w - 2w + 16#

Now, we can factor by grouping like this:
#(3w^2 - 24w)# and #(-2w + 16)#

Technically, we just separated the expression into two parts.
Now we factor each part:
#3w(w - 8)# and #-2(w - 8)#

Notice how there is #(w-8)# in both parts? Now we can put them together like this!:
#(3w-2)(w-8)#

So our final expression factored with the #w^4# in the beginning is:
#w^4(3w-2)(w-8)#

Let's check our work to make sure that this is equivalent to the original expression #3w^6 - 26w^5 + 16w^4# by distributing:
#w^4(3w-2)(w-8)#

#(3w^5 - 2w^4)(w-8)#

#3w^6 - 24w^5 - 2w^5 + 16w^4#

#3w^6 - 26w^5 + 16w^4#

Yayyy it matches the original expression; therefore the factored expression is #w^4(3w-2)(w-8)#.

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