Exercise 5.2
Question 1(a)
Expand:
\[ (a+b+2c)^2 \]
Solution:
Using the identity
\[ (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx \]
Here,
\[ x=a,\qquad y=b,\qquad z=2c \]
Substitute these values into the identity:
\[ \begin{aligned} (a+b+2c)^2 &=a^2+b^2+(2c)^2+2(a)(b)+2(b)(2c)+2(a)(2c)\\ &=a^2+b^2+4c^2+2ab+4bc+4ac \end{aligned} \]
Final Answer:
\[ \boxed{a^2+b^2+4c^2+2ab+4ac+4bc} \]
Exercise 5.2
Question 1(b)
Expand:
\[ (3a-2b+c)^2 \]
Solution:
Using the identity
\[ (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx \]
Let
\[ x=3a,\qquad y=-2b,\qquad z=c \]
Substituting these values,
\[
\begin{aligned}
(3a-2b+c)^2
&=(3a)^2+(-2b)^2+c^2\\
&\quad+2(3a)(-2b)+2(-2b)(c)+2(3a)(c)
&=9a^2+4b^2+c^2-12ab-4bc+6ac
\end{aligned}
\]
Final Answer:
\[ \boxed{9a^2+4b^2+c^2-12ab+6ac-4bc} \]
Exercise 5.2
Question 1(c)
Expand:
\[ (-2a+b+c)^2 \]
Solution:
Using the identity
\[ (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx \]
Let
\[ x=-2a,\qquad y=b,\qquad z=c \]
Substituting these values,
\[
\begin{aligned}
(-2a+b+c)^2
&=(-2a)^2+b^2+c^2\\
&\quad+2(-2a)(b)+2(b)(c)+2(-2a)(c)
&=4a^2+b^2+c^2-4ab+2bc-4ac
\end{aligned}
\]
Final Answer:
\[ \boxed{4a^2+b^2+c^2-4ab-4ac+2bc} \]
Exercise 5.2
Question 1(d)
Expand:
\[ (-3x+y-2z)^2 \]
Solution:
\[ \begin{aligned} (-3x+y-2z)^2 &=(-3x)^2+y^2+(-2z)^2\\ &\quad+2(-3x)(y)+2(y)(-2z)+2(-3x)(-2z)\\ &=9x^2+y^2+4z^2-6xy-4yz+12xz \end{aligned} \]
Answer:
\[ \boxed{9x^2+y^2+4z^2-6xy+12xz-4yz} \]
Question 1(e)
Expand:
\[ (m+5p-2n)^2 \]
Solution:
\[ \begin{aligned} (m+5p-2n)^2 &=m^2+(5p)^2+(-2n)^2\\ &\quad+2(m)(5p)+2(5p)(-2n)+2(m)(-2n)\\ &=m^2+25p^2+4n^2+10mp-20pn-4mn \end{aligned} \]
Answer:
\[ \boxed{m^2+25p^2+4n^2+10mp-4mn-20pn} \]
Question 1(f)
Expand:
\[ (a^2+b^2+c^2)^2 \]
Solution:
\[ \begin{aligned} (a^2+b^2+c^2)^2 &=(a^2)^2+(b^2)^2+(c^2)^2\\ &\quad+2(a^2)(b^2)+2(b^2)(c^2)+2(c^2)(a^2)\\ &=a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2 \end{aligned} \]
Answer:
\[ \boxed{a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2} \]
Question 1(g)
Expand:
\[ (ab+bc+ca)^2 \]
Solution:
\[ \begin{aligned} (ab+bc+ca)^2 &=(ab)^2+(bc)^2+(ca)^2\\ &\quad+2(ab)(bc)+2(bc)(ca)+2(ca)(ab)\\ &=a^2b^2+b^2c^2+c^2a^2+2ab^2c+2abc^2+2a^2bc \end{aligned} \]
Answer:
\[ \boxed{a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2} \]
Question 1(h)
Expand:
\[ \left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)^2 \]
Solution:
\[ \begin{aligned} \left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)^2 &=\frac{x^2}{y^2}+\frac{y^2}{z^2}+\frac{z^2}{x^2}\\ &\quad+2\left(\frac{x}{y}\cdot\frac{y}{z}\right) +2\left(\frac{y}{z}\cdot\frac{z}{x}\right) +2\left(\frac{z}{x}\cdot\frac{x}{y}\right)\\ &=\frac{x^2}{y^2}+\frac{y^2}{z^2}+\frac{z^2}{x^2} +\frac{2x}{z}+\frac{2y}{x}+\frac{2z}{y} \end{aligned} \]
Answer:
\[ \boxed{\frac{x^2}{y^2}+\frac{y^2}{z^2}+\frac{z^2}{x^2}+\frac{2x}{z}+\frac{2y}{x}+\frac{2z}{y}} \]
Question 1(i)
Expand:
\[ \left(\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\right)^2 \]
Solution:
\[ \begin{aligned} \left(\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\right)^2 &=\frac{x^2}{y^2z^2} +\frac{y^2}{z^2x^2} +\frac{z^2}{x^2y^2}\\ &\quad+2\left(\frac{x}{yz}\cdot\frac{y}{zx}\right) +2\left(\frac{y}{zx}\cdot\frac{z}{xy}\right) +2\left(\frac{z}{xy}\cdot\frac{x}{yz}\right)\\ &=\frac{x^2}{y^2z^2} +\frac{y^2}{z^2x^2} +\frac{z^2}{x^2y^2} +\frac{2}{z^2} +\frac{2}{x^2} +\frac{2}{y^2} \end{aligned} \]
Answer:
\[ \boxed{ \frac{x^2}{y^2z^2} +\frac{y^2}{z^2x^2} +\frac{z^2}{x^2y^2} +\frac{2}{x^2} +\frac{2}{y^2} +\frac{2}{z^2} } \]