Question #8ae1f

3 Answers
Douglas K.
Oct 25, 2017

Because the leading coefficient is 1 we know that the form is:

#(y" ? "r_1)(y" ? "r_2)#

where ? is either a plus or a minus .

Because the last term is +21 we know that both ?s are the same (either both plus or both minus).

Because the middle term is -22 we know that both ?s are minus:

#(y-r_1)(y-r_2)#

We know that #(-r_1)(-r_2) = 21# and #-r_1-r_2 = -22#

The numbers 1 and 21 will work:

#(y-1)(y-21)#

Sunayan S
Oct 25, 2017

See the answer below...

See Connie's answer for better experience...

Explanation:

#y^2-22y+21#
#=y^2-21y-y+21# [By middle term Factor]
#=y(y-21)-1(y-21)#
#=(y-1)(y-21)#

Hope it helps...
Thank you..

Connie
Oct 25, 2017

This one is really easy, even tho' it may not look like it.

Explanation:

Here's the factorization:

#(y-1) * (y-21)#

(It makes NO difference that the variable here is "y" and not "x".)

To factorize a polynomial, you need to ask, "What two numbers MULTIPLY to the LAST TERM #(+ 21)# and ADD to the MIDDLE TERM #(-22)#?

Since a negative times a negative equals a positive, both of the constants in the factorization will be negative: #(- 1) times (- 21) = (+ 21)#.

Also, #(- 1) + (- 21)# ADD to #(-22)#, for the coefficient of the middle term.

If there are more than two possible factors, write them out and try different combinations. Start with the constant - the last term. Example:

#x^2 + 11x +24#

What two numbers can you MULTIPLY to get 24?
#1 * 24#
#2 * 12#
#3 * 8#
#4 * 6#

Which of those combinations ADDS to #11#?
#3 + 8# does, so this factors out to:
#(x+3)*(x+8)#

How about this one:
#x^2 + 10x +24#?

What two factors ADD to 10?
#4 and 6#, so this factors out to:
#(x+4)*(x+6)#

Going back to your example, let's FOIL the terms to get the polynomial back:

FOIL stands for "first", "outer", "inner" and "last", referring to the the terms in the expression.

You begin by Multiplying the First Terms : #y * y = y^2#

then Multiply the Outer Terms : #y * (-21) = -21y#

then Multiply the Inner Terms : #(-1) * y = -y#

and finally, Multiply the Last Terms: #(-1) * (-21) = +21#.

Now you're going to Add like terms -- the y-squared terms, the y-terms and the constants. Write them in columns:

#y^2 + (-21y)#
#.... +(- 1y) ...+ 21#

(The dots or periods above are only to keep the terms in their proper column.)

The answer is (of course):

y^2 - 22y + 21

Comment if you want or need more explanation.
Connie

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