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Exercise Set D
Question 1
Expand the following:
\begin{align}(x+2)^5\end{align}
Answer
\(x^5+10x^4+40x^3+80x^2+80x+32\)
Solution
\begin{align}(x+2)^5 &= \binom{5}{0}x^5(2^0) +\binom{5}{1}x^4(2^1)+\binom{5}{2}x^3(2^2)+\binom{5}{3}x^2(2^3)+\binom{5}{4}x^1(2^4)+\binom{5}{5}x^0(2^5)\\&=(1)x^5(1)+(5)x^4(2)+(10)x^3(4)+(10)x^2(8)+(5)x(16)+(1)(1)(32)\\&=x^5+10x^4+40x^3+80x^2+80x+32 \end{align}
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Question 2
Expand the following:
\begin{align}(x+y)^3\end{align}
Answer
\(x^3+3x^2y+3xy^2+y^3\)
Solution
\begin{align}(x+y)^3 &= \binom{3}{0}x^3(y^0) +\binom{3}{1}x^2(y^1)+\binom{3}{2}x^1(y^2)+\binom{3}{3}x^0(y^3)\\&=(1)x^3(1)+(3)x^2(y)+(3)x(y^2)+(1)(1)(y^3)\\&=x^3+3x^2y+3xy^2+y^3 \end{align}
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Question 3
Expand the following:
\begin{align}(x-6)^4\end{align}
Answer
\(x^4-24x^3+216x^2-864x+1296\)
Solution
\begin{align}(x-6)^4 &= \binom{4}{0}x^4(-6)^0 +\binom{4}{1}x^3(-6)^1+\binom{4}{2}x^2(-6)^2+\binom{4}{3}x^1(-6)^3+\binom{4}{4}x^0(-6)^4\\&=(1)x^4(1)+(4)x^3(-6)+(6)x^2(36)+(4)x(-216)+(1)(1)(1296)\\&=x^4-24x^3+216x^2-864x+1296 \end{align}
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Question 4
Expand the following:
\begin{align}(2x+5)^3\end{align}
Answer
\(8x^3+60x^2+150x+125\)
Solution
\begin{align}(2x+5)^3 &= \binom{3}{0}(2x)^3(5^0) +\binom{3}{1}(2x)^2(5^1)+\binom{3}{2}(2x)^1(5^2)+\binom{3}{3}(2x)^0(5^3)\\&=(1)(8x^3)(1)+(3)(4x^2)(5)+(3)(2x)(25)+(1)(1)(125)\\&=8x^3+60x^2+150x+125 \end{align}
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Question 5
Expand the following:
\begin{align}(2x-3y)^4\end{align}
Answer
\(16x^4-96x^3y+216x^2y^2-216xy^3+81y^4\)
Solution
\begin{align}(2x-3y)^4 &= \binom{4}{0}(2x)^4(-3y)^0 +\binom{4}{1}(2x)^3(-3y)^1+\binom{4}{2}(2x)^2(-3y)^2+\binom{4}{3}(2x)^1(-3y)^3+\binom{4}{4}(2x)^0(-3y)^4\\&=(1)(16x^4)(1)+(4)(8x^3)(-3y)+(6)(4x^2)(9y^2)+(4)(2x)(-27y^3)+(1)(1)(81y^4)\\&=16x^4-96x^3y+216x^2y^2-216xy^3+81y^4 \end{align}
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