Question

If \[a + b + c = 1\], \[{a^2} + {b^2} + {c^2} = 21\], \[abc = 8\] . Find the value of \[\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)\] .

Answer 1

Hint: In this question, we have three algebraic equations.
We have been asked to find out the value of the given expression.
We have to first multiply the factors with each other. Then putting the given values from the equations given, we will get a shorter form. After that we will use one algebraic formula and again putting the values, we will get the required value.

Formula used: Algebraic formula,
\[{\left( {a + b + c} \right)^2} = \left( {{a^2} + {b^2} + {c^2}} \right) + 2ab + 2bc + 2ac\]

Complete step-by-step solution:
It is given that, \[a + b + c = 1\], \[{a^2} + {b^2} + {c^2} = 21\], \[abc = 8\].
We need to find out the value of \[\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)\].
Here we have,
\[a + b + c = 1\] ……………...… (i)
\[{a^2} + {b^2} + {c^2} = 21\] ……………..… (ii)
\[abc = 8\] …………..…. (iii)
Now the required expression is,
\[\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)\]
Multiplying the last two factors,
\[ \Rightarrow \left( {1 - a} \right)\left( {1 - c - b + bc} \right)\]
Multiplying all the factors,
\[ \Rightarrow 1 - c - b + bc - a + ac + ab - abc\]
Putting the values from (iii),
\[ \Rightarrow 1 - \left( {a + b + c} \right) + ab + bc + ac - 8\]
Putting the values from (i),
\[ \Rightarrow 1 - 1 + ab + bc + ac - 8\]
Simplifying we get,
\[ \Rightarrow \left( {ab + bc + ac} \right) - 8\]…………... (iv)
Again, we have, \[a + b + c = 1\]
Squaring both the sides we get,
\[{\left( {a + b + c} \right)^2} = {1^2}\]
\[\left( {{a^2} + {b^2} + {c^2}} \right) + 2ab + 2bc + 2ac = 1\]
Using the algebraic formula, \[{\left( {a + b + c} \right)^2} = \left( {{a^2} + {b^2} + {c^2}} \right) + 2ab + 2bc + 2ac\]
Putting the values from (ii) we get,
$\Rightarrow$\[21 + 2\left( {ab + bc + ac} \right) = 1\]
Rearranging the terms, we get,
$\Rightarrow$\[ab + bc + ac = \dfrac{{1 - 21}}{2}\]
Subtracting the terms, we get,
$\Rightarrow$\[ab + bc + ca = - \dfrac{{20}}{2}\]
Hence,
\[ab + bc + ac = - 10\]
Hence putting the values of \[ab + bc + ac = - 10\] in (iv), we get,
$\Rightarrow$\[\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)\]
\[ \Rightarrow - 10 - 8 = - 18\]
Thus, \[\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right) = - 18\]

Hence, the value of \[\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)\] is \[ - 18\].

Note: Multiplication of two Binomials:
Suppose $\left( {a + b} \right)$ and $\left( {c + d} \right)$ are two binomials. By using the distributive law of multiplication, we may find their product as given below,
\[\left( {a + b} \right) \times \left( {c + d} \right)\]
First, we will multiply with each term on both brackets,
\[ \Rightarrow a \times \left( {c + d} \right) + b \times \left( {c + d} \right)\]
Simplifying we get,
\[ \Rightarrow \left( {a \times c + a \times d} \right) + \left( {b \times c + b \times d} \right)\]
Hence,
\[ \Rightarrow ac + ad + bc + bd\]
Algebraic expression:
In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations.
For example,
\[{x^2} + 6xy + 7\] is an algebraic expression where \[7\] is the integer constants and x and y are the variables, + is the algebraic operation.