 |
| a | $\begin{align}&(x+3)^3=x^3+3\times x^2\times 3+3\times x \times 3^2+3^3\\[5pt]
&=\underline{\underline{x^3+9x^2+27x+27}}
\end{align}$ |
| b | $\begin{align}&(y+2)^3=y^3+3\times y^2\times 2+3\times y \times 2^2+2^3\\[5pt]
&=\underline{\underline{y^3+6y^2+12y+8}}
\end{align}$ |
| c | $\begin{align}&(a-5)^3\\[5pt]&=(a+(-5))^2\\[5pt]&=a^3+3\times a^2\times(-5)+3\times a\times(-5)^2+(-5)^3\\[5pt]
&=\underline{\underline{a^3-15a^2+75a-125}}
\end{align}$ |
| d | $\begin{align}&(3+t)^3=3^3+3\times 3^2\times t+3\times 3 \times t^2+t^3\\[5pt]
&=\underline{\underline{27+27t+9t^2+t^3}}
\end{align}$ |
| e | $\begin{align}&(x-2)^3\\[5pt]&=(x+(-2))^2\\[5pt]&=x^3+3\times x^2\times(-2)+3\times x\times(-2)^2+(-2)^3\\[5pt]
&=\underline{\underline{x^3-6x^2+12x-8}}
\end{align}$ |
 |
| a | $\begin{align}&(m+2)^3=m^3+3\times m^2\times 2+3\times m \times 2^2+2^3\\[5pt]
&=\underline{\underline{m^3+6m^2+12m+8}}
\end{align}$ |
| b | $\begin{align}&(x+4)^3=x^3+3\times x^2\times 4+3\times x \times 4^2+4^3\\[5pt]
&=\underline{\underline{x^3+12x^2+48x+64}}
\end{align}$ |
| c | $\begin{align}&(b-2)^3=b^3-3\times b^2\times 2+3\times b \times 2^2-2^3\\[5pt]
&=\underline{\underline{b^3-6b^2+12b-8}}
\end{align}$ |
| d | $\begin{align}&(t-10)^3=t^3-3\times t^2\times 10+3\times t \times 10^2-10^3\\[5pt]
&=\underline{\underline{t^3-30t^2+300t-1000}}
\end{align}$ |
| e | $\begin{align}&(5+p)^3=5^3+3\times 5^2\times p+3\times 5 \times p^2+p^3\\[5pt]
&=\underline{\underline{125+75p+15p^2+p^3}}
\end{align}$ |
| f | $\begin{align}&(6+k)^3=6^3+3\times 6^2\times k+3\times 6 \times k^2+k^3\\[5pt]
&=\underline{\underline{216+108k+18k^2+k^3}}
\end{align}$ |
| g | $\begin{align}&(1+b)^3=1^3+3\times 1^2\times b+3\times 1 \times b^2+b^3\\[5pt]
&=\underline{\underline{1+3b+3b^2+b^3}}
\end{align}$ |
| h | $\begin{align}&(4-x)^3=4^3-3\times 4^2\times x+3\times 4 \times x^2-x^3\\[5pt]
&=\underline{\underline{64-48x+12x^2-x^3}}
\end{align}$ |
| i | $\begin{align}&(2-p)^3=2^3-3\times 2^2\times p+3\times 2 \times p^2-p^3\\[5pt]
&=\underline{\underline{8-12p+6p^2-p^3}}
\end{align}$ |
| j | $\begin{align}&(9-t)^3=9^3-3\times 9^2\times t+3\times 9 \times t^2-t^3\\[5pt]
&=\underline{\underline{729-243t+27t^2-t^3}}
\end{align}$ |
| k | $\begin{align}&(-m+3)^3=(-m)^3+3\times (-m)^2\times 3+3\times (-m) \times 3^2+3^3\\[5pt]
&=\underline{\underline{-m^3+9m^2-27m+27}}
\end{align}$ |
| l | $\begin{align}&(-5-y)^3=(-5)^3+3\times (-5)^2\times (-y)+3\times (-5) \times (-y)^2+(-y)^3\\[5pt]
&=\underline{\underline{-125-75y-15y^2-y^3}}
\end{align}$ |
| m | $\begin{align}&(ab+c)^3=(ab)^3+3\times (ab)^2\times c+3\times (ab) \times c^2+c^3\\[5pt]
&=\underline{\underline{a^3b^3+3a^2b^2c+3abc^2+c^3}}
\end{align}$ |
| n | $\begin{align}&(2x+3y)^3=(2x)^3+3\times (2x)^2\times (3y)+3\times (2x) \times (3y)^2+(3y)^3\\[5pt]
&=\underline{\underline{8x^3+36x^2y+54xy^2+27y^3}}
\end{align}$ |
| o | $\begin{align}&(3x+4y)^3=(3x)^3+3\times (3x)^2\times (4y)+3\times (3x) \times (4y)^2+(4y)^3\\[5pt]&=27x^3+3\times (9x^2)\times (4y)+3\times (3x) \times (16y^2)+(64y^3)\\[5pt]
&=\underline{\underline{27x^3+108x^2y+144xy^2+64y^3}}
\end{align}$ |
| p | $\begin{align}&(2a-5b)^3=(2a)^3-3\times (2a)^2\times (5b)+3\times (2a) \times (5b)^2-(5b)^3\\&=8a^3-3\times (4a)^2\times (5b)+3\times (2a) \times (25b)^2-(125b)^3\\
&=\underline{\underline{8a^3-60a^2b+150ab^2-125b^3}}
\end{align}$ |
 |
| a | $\begin{align}&a^3+3a^2b+3ab^2+b^3\\[5pt]
&=\underline{\underline{(a+b)^3}}
\end{align}$ |
| b | $\begin{align}&c^3-3c^2d+3cd^2-d^3\\[5pt]
&=\underline{\underline{(c-d)^3}}
\end{align}$ |
| c | $\begin{align}&x^3+6x^2+12x+8\\[5pt]&=x^3+3\times x^2\times2+3\times x\times 4+8\\[5pt]&=x^3+3\times x^2\times2+3\times x\times 2^2+2^3\\[5pt]
&=\underline{\underline{(x+2)^3}}
\end{align}$ |
| d | $\begin{align}&y^3-18y^2+108y-216\\[5pt]&=y^3-3\times y^2\times6+3\times y\times 36-216\\[5pt]&=y^3-3\times y^2\times6+3\times y\times 6^2-6^3\\[5pt]
&=\underline{\underline{(y-6)^3}}
\end{align}$ |
| e | $\begin{align}&1+3x+3x^2+x^3\\[5pt]&=1^3+3\times 1^2\times x+3\times 1\times x^2+x^3\\[5pt]
&=\underline{\underline{(1+x)^3}}
\end{align}$ |
| f | $\begin{align}&64-48x+12x^2-x^3\\[5pt]&=64-3\times 16\times x+3\times 4\times x^2-x^3\\[5pt]&=4^3-3\times 4^2\times x+3\times 4\times x^2-x^3\\[5pt]
&=\underline{\underline{(4-x)^3}}
\end{align}$ |
 |
| a | $\begin{align}&(a+5)^3\\[5pt]&=a^3+3\times a^2\times 5+3\times a \times 5^2+5^3\\[5pt]
&=\underline{\underline{a^3+15a^2+75a+125}}
\end{align}$ |
 |
i
x=2
| $\begin{align}&(x+3)^3=x^3+3\times x^2\times 3+3\times x\times 3^2+3^3\\[5pt]&x=2 , \text{Left
side}=(2+3)^3\\[5pt]&=(5)^3\\[5pt]&=\underline{\underline{125}}
\end{align}$ |
| $\begin{align}&\text{Right side}=2^3+3\times 2^2\times 3+3\times 2\times 3^2+3^3\\[5pt]&=2^3+9\times2^2+27\times 2+27\\[5pt]&=8+36+54+27\\[5pt]&=\underline{\underline{125}}
\end{align}$ |
ii
x=4 | $\begin{align}&(x+3)^3=x^3+3\times x^2\times 3+3\times x\times 3^2+3^3\\[5pt]&x=4 , \text{Left side}=(4+3)^3\\[5pt]&=(7)^3\\[5pt]&=\underline{\underline{343}}
\end{align}$ |
| $\begin{align}&\text{Right side}=4^3+3\times 4^2\times 3+3\times 4\times 3^2+3^3\\[5pt]&=64+3\times16\times 3+3\times 4\times 9+27\\[5pt]&=64+144+108+27\\[5pt]&=\underline{\underline{343}}
\end{align}$
$\begin{align}\\[5pt]\text{Both sides values are equal}
\end{align}$ |
 |
| i | $\begin{align}&64-3\times16\times3+3\times4\times9-27\\[5pt]&=4^3-3\times4^2\times3+3\times4\times3^2-3^3\\[5pt]&=(4-3)^3\\[5pt]&=(1)^3\\[5pt]
&=\underline{\underline{1}}
\end{align}$ |
| ii | $\begin{align}&216-3\times36\times5+3\times6\times25-125\\[5pt]&=6^3-3\times6^2\times5+3\times6\times5^2-5^3\\[5pt]&=(6-5)^3\\[5pt]&=(1)^3\\[5pt]
&=\underline{\underline{1}}
\end{align}$ |
 |
| a | $\begin{align}&21^3\\[5pt]&=(20+1)^3\\[5pt]&=20^3+3\times20^2\times 1+3\times 20\times1^2+1^3\\[5pt]&=8000+3\times400+60+1\\[5pt]&=8000+1200+60+1\\[5pt]
&=\underline{\underline{9261}}
\end{align}$ |
| b | $\begin{align}&102^3\\[5pt]
&=(100+2)^3\\[5pt]
&=100^3+3\times100^2\times 2+3\times 100\times2^2+2^3\\[5pt]
&=1000000+3\times10000\times 2+3\times 100\times 4+8\\[5pt]
&=1000000+60000+1200+8\\[5pt]
&=\underline{\underline{1061208}}
\end{align}$ |
| c | $\begin{align}&17^3\\[5pt]&=(20-3)^3\\[5pt]
&=20^3-3\times20^2\times 3+3\times 20\times3^2-3^3\\[5pt]
&=8000-3\times400\times 3+3\times 20\times 9-27\\[5pt]
&=8000-3600+540-27\\[5pt]
&=\underline{\underline{4913}}
\end{align}$ |
| d | $\begin{align}&98^3\\[5pt]&=(100-2)^3\\[5pt]
&=100^3-3\times 100^2\times 2+3\times 100\times2^2-2^3\\[5pt]
&=1000000-3\times10000\times 2+3\times 100\times 4-8\\[5pt]
&=1000000-60000+1200-8\\[5pt]
&=\underline{\underline{941192}}
\end{align}$ |
 |
| a | $\begin{align}&(2a-5)^3\\[5pt]&=(2a)^3-3\times(2a)^2\times 5+3\times (2a)\times5^2-5^3\\[5pt]&=8a^3-3\times4a^2\times 5+3\times(2a)\times 25-125\\[5pt]
&=\underline{\underline{8a^3-60a^2+150a-125 cm^3}}
\end{align}$ |
 |
| a | $\begin{align}&x^3-3x^2y+3xy^2-y^3=(x-y)^3\\[5pt]&x=25,y=23\\[5pt]&=25^3-3\times25^2\times 23+3\times25\times 23^2-23^3\\[5pt]&=(25-23)^3\\[5pt]&=(2)^3\\[5pt]
&=\underline{\underline{8}}
\end{align}$ |
By studying this lesson, you will be able to
Expand the cube (third power) of a binomial expression.