Several algebra questions regarding factorisation

Jun 2010
54
0
A very good day to all.
I have several questions all relating to algebra, and I will be very thankful if you can help me solve them. Please also show your workings clearly so that I may understand wholly. I am sorry if these questions take a lot a time.

Factorise the following:

1.\(\displaystyle (x + y)(a + b) - (y + z)(a + b)\)


2. \(\displaystyle (2x + y)^2 -3(2x + y)\)


3. \(\displaystyle 5(m - 2n) - (m -2n)^2\)


4. \(\displaystyle p^2 -9q^2 +6qr -r^2\)


5. \(\displaystyle 4a^2 -b^2 -2bc -c^2\)
 

Chris L T521

MHF Hall of Fame
May 2008
2,844
2,046
Chicago, IL
A very good day to all.
I have several questions all relating to algebra, and I will be very thankful if you can help me solve them. Please also show your workings clearly so that I may understand wholly. I am sorry if these questions take a lot a time.

Factorise the following:

1.\(\displaystyle (x + y)(a + b) - (y + z)(a + b)\)
Observe that \(\displaystyle (a+b)\) is the common factor. So \(\displaystyle (x+y)(a+b)-(y+z)(a+b)=(a+b)(\ldots)\)

2. \(\displaystyle (2x + y)^2 -3(2x + y)\)
Observe that \(\displaystyle (2x+y)\) is the common factor. So \(\displaystyle (2x+y)^2-3(2x+y)=(2x+y)(\ldots)\)

3. \(\displaystyle 5(m - 2n) - (m -2n)^2\)
Observe that \(\displaystyle (m-2n)\) is the common factor. So \(\displaystyle 5(m-2n)-(m-2n)^2=(m-2n)(\ldots)\)

4. \(\displaystyle p^2 -9q^2 +6qr -r^2\)
This one is slightly more complicated. First observe that \(\displaystyle p^2-9q^2+6rq-r^2=p^2-(9q^2-6qr+r^2)\).

Now, \(\displaystyle 9q^2-6qr+r^2=(3q-r)^2\). So now we are left with \(\displaystyle p^2-(3q-r)^2\), which can be factored using the difference of squares formula.


5. \(\displaystyle 4a^2 -b^2 -2bc -c^2\)
This is similar to #4. Observe that \(\displaystyle 4a^2-b^2-2bc-c^2=(2a)^2-(b^2+2bc+c^2)\).

Now, \(\displaystyle b^2+2bc+c^2=(b+c)^2\). So now we are left with \(\displaystyle (2a)^2-(b+c)^2\), which can be factored using the difference of squares formula.

Does this make sense? Can you try these problems now?
 
Jun 2010
54
0
Thank you so much for your help Chris!

However, I still could not work out questions 1, 2 and 3 myself. I know the common factors but I am not sure about the other terms.

Also, I would appreciate if you could verify my answers for question 4 and 5.
The answer for question 4 might be---

\(\displaystyle p^2 -(3q -r)^2 \)
\(\displaystyle =[p^2 +(3q-r)][p^2 -(3q -r)]\)
\(\displaystyle =(p^2 +3q -r)(p^2 -3q +r)\)

Whereas for question 5---

\(\displaystyle (2a)^2 -(b+c)^2\)
\(\displaystyle =[2a +(b+c)][2a -(b +c)\)
\(\displaystyle = (2a +b +c)(2a -b -c)\)

Is my answers and workings correct?
 
Jul 2010
456
138
However, I still could not work out questions 1, 2 and 3 myself. I know the common factors but I am not sure about the other terms.

\(\displaystyle (a+b)(x+y-y-z)=(a+b)(x-z)\)


\(\displaystyle (2x+y)^2-3(2x+y)=(2x+y)(2x+y)-3(2x+y)=(2x+y)(2x+y-3)\)


same as second :D
\(\displaystyle (m-2n)(5-m+2n)\)