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Algebraic Expressions and Identities | Exercise 9.3

Question 10

Q5) Add: p\left(p-q\right),\ q\left(q-r\right)\ and\ r\left(r-p\right)

(b) Add: 2x\left(z-x-y\right)and\ 2y\left(z-y-zx\right)

(c) Subtract: 3l\left(l-4m+5n\right)from\ 4l\left(10n-3m+2l\right)

(d) Subtract: 3a\left(a+b+c\right)-2b\left(a-b+c\right)from\ 4c\left(-a+b+c\right)

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Solution:

(a) p\left(p-q\right)+q\left(q-r\right)+r\left(r-p\right)

= p^2-pq+q^2-qr+r^2-rp

= p^2+q^2+r^2-pq-qr-rp

(b) 2x(z-x-y)+2y(z-y-z)

= 2xz-2x^2-2xy+2yz-2y^2-2xy

= 2xz-2xy-2xy+2yz-2x^2-2y^2

= -2x^2-2y^2-4xy+2yz+2zx

(c) 4l\left(10n-3m+2l\right)-3l\left(l-4m+5n\right)

= 40\ln-12lm+8l^2-3l^2+12lm-15l\ n

= 8l^2-3l^2-12lm+12lm+40\ln-15\ln

= 5l^2+25\ln

(d) 4c\left(-a+b+c\right)\ -\left[3a\left(a+b+c\right)-2b\left(a-b+c\right)\right]

= -4ac+4bc+4c^2-\left[3a^2+3ab+3ac-2ab+2b^2-2bc\right]

= -4ac+4bc+4c^2-\left[3a^2+2b^2+3ab-2bc+3ac-2ab\right]

= -4ac+4bc+4c^2-\left[3a^2+2b^2+ab+3ac-2bc\right]

= -4ac+4bc+4c^2-3a^2-2b^2-ab-3ac+2bc

= -3a^2-2b^2+4c^2-ab+4bc+2bc-4ac-3ac

= -3a^2-2b^2+4c^2-ab+6bc-7ac

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Question 10

Q5) Add: p\left(p-q\right),\ q\left(q-r\right)\ and\ r\left(r-p\right)

(b) Add: 2x\left(z-x-y\right)and\ 2y\left(z-y-zx\right)

(c) Subtract: 3l\left(l-4m+5n\right)from\ 4l\left(10n-3m+2l\right)

(d) Subtract: 3a\left(a+b+c\right)-2b\left(a-b+c\right)from\ 4c\left(-a+b+c\right)

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Solution:

(a) p\left(p-q\right)+q\left(q-r\right)+r\left(r-p\right)

= p^2-pq+q^2-qr+r^2-rp

= p^2+q^2+r^2-pq-qr-rp

(b) 2x(z-x-y)+2y(z-y-z)

= 2xz-2x^2-2xy+2yz-2y^2-2xy

= 2xz-2xy-2xy+2yz-2x^2-2y^2

= -2x^2-2y^2-4xy+2yz+2zx

(c) 4l\left(10n-3m+2l\right)-3l\left(l-4m+5n\right)

= 40\ln-12lm+8l^2-3l^2+12lm-15l\ n

= 8l^2-3l^2-12lm+12lm+40\ln-15\ln

= 5l^2+25\ln

(d) 4c\left(-a+b+c\right)\ -\left[3a\left(a+b+c\right)-2b\left(a-b+c\right)\right]

= -4ac+4bc+4c^2-\left[3a^2+3ab+3ac-2ab+2b^2-2bc\right]

= -4ac+4bc+4c^2-\left[3a^2+2b^2+3ab-2bc+3ac-2ab\right]

= -4ac+4bc+4c^2-\left[3a^2+2b^2+ab+3ac-2bc\right]

= -4ac+4bc+4c^2-3a^2-2b^2-ab-3ac+2bc

= -3a^2-2b^2+4c^2-ab+4bc+2bc-4ac-3ac

= -3a^2-2b^2+4c^2-ab+6bc-7ac

Our top 5% students will be awarded a special scholarship to Lido.

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