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Math Help - Several algebra questions regarding factorisation

  1. #1
    Junior Member PythagorasNeophyte's Avatar
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    Several algebra questions regarding factorisation

    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.  (x + y)(a + b) - (y + z)(a + b)


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


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


    4. p^2 -9q^2 +6qr -r^2


    5. 4a^2 -b^2 -2bc -c^2
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by PythagorasNeophyte View Post
    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.  (x + y)(a + b) - (y + z)(a + b)
    Observe that (a+b) is the common factor. So (x+y)(a+b)-(y+z)(a+b)=(a+b)(\ldots)

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

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

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

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


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

    Now, b^2+2bc+c^2=(b+c)^2. So now we are left with (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?
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  3. #3
    Junior Member PythagorasNeophyte's Avatar
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    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---

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

    Whereas for question 5---

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

    Is my answers and workings correct?
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  4. #4
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by PythagorasNeophyte View Post
    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.
    1
    (a+b)(x+y-y-z)=(a+b)(x-z)

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

    3
    same as second
    (m-2n)(5-m+2n)
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