Question

How do you factor \[3{b^2} - 48b + 189\] ?

Answer 1

Hint: The given expression is a polynomial of degree 2. Instead of ‘x’ as a variable we have ‘b’ as a variable. We can solve this by using factorization methods or by using quadratic formulas. We use quadratic formula if factorization fails. We know that a polynomial equation has exactly as many roots as its degree.

Complete step-by-step answer:
The degree of the equation \[3{b^2} - 48b + 189\] is 2, so the number of roots of the given equation is 2.
On comparing the given equation with the standard quadratic equation
\[A{b^2} + Bb + C = 0\] .
Where ‘A’ and ‘B’ are coefficients of \[{b^2}\] and coefficient of ‘b’ respectively.
We have \[A = 3\] , \[B = - 48\] and \[C = 189\] .
The standard form of the factorization of quadratic equation is
\[A{b^2} + {B_1}b + {B_2}b + C = 0\] , which satisfies the condition \[{B_1} \times {B_2} = A \times C\] and \[{B_1} + {B_2} = B\] .
We can write the given equation as
\[3{b^2} - 27b - 21b + 189\] ,
where \[{B_1} = - 27\] and \[{B_2} = - 21\] . Also \[{B_1} \times {B_2} = ( - 27) \times ( - 21) = 567(A \times C)\] and \[{B_1} + {B_2} = ( - 27) + ( - 21) = - 48(B)\] .
 \[ \Rightarrow 3{b^2} - 48b + 189 = 3{b^2} - 27b - 21b + 189\]
 \[ = 3{b^2} - 27b - 21b + 189\]
In the first two terms we take ‘3b’ as common and in the remaining term we take -21 as common,
 \[ = 3b(b - 9) - 21(b - 9)\]
Again taking \[(b - 9)\] as common we have,
 \[ = (b - 9)(3b - 21)\]
Hence, the factors of \[3{b^2} - 48b + 189\] are \[(b - 9)\] and \[(3b - 21)\] .

Note: We can find the zeros or roots of the given quadratic expression by equating the obtained factors to zero. That is,
 \[(b - 9)(3b - 21) = 0\] .
By zero multiplication property we have,
 \[(b - 9) = 0\] and \[(3b - 21) = 0\] .
 \[b = 9\] and \[3b = 21\] .
 \[b = 9\] and \[b = \dfrac{{21}}{3}\] .
 \[ \Rightarrow b = 9\] and \[b = 7\] .
These are the roots of the given problem.
In various fields of mathematics require the point at which the value of a polynomial is zero, those values are called the factors/solution/zeros of the given polynomial. On the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts.