Answer :
To factor the polynomial
[tex]$$3x^2 - 18x + 27,$$[/tex]
we proceed as follows:
1. Factor out the greatest common factor (GCF):
All the coefficients (3, -18, 27) have a common factor of 3. Thus, we factor 3 out of the polynomial:
[tex]$$3x^2 - 18x + 27 = 3(x^2 - 6x + 9).$$[/tex]
2. Factor the quadratic inside the parentheses:
Next, we need to factor the quadratic expression
[tex]$$x^2 - 6x + 9.$$[/tex]
Look for two numbers that multiply to [tex]$9$[/tex] (the constant term) and add to [tex]$-6$[/tex] (the coefficient of [tex]$x$[/tex]). The numbers [tex]$-3$[/tex] and [tex]$-3$[/tex] satisfy these conditions since:
[tex]$$(-3) \times (-3) = 9$$[/tex]
and
[tex]$$(-3) + (-3) = -6.$$[/tex]
Thus, the quadratic factors nicely as:
[tex]$$x^2 - 6x + 9 = (x-3)(x-3) = (x-3)^2.$$[/tex]
3. Write the complete factorization:
Replace the factored quadratic back into the expression:
[tex]$$3x^2 - 18x + 27 = 3(x-3)^2.$$[/tex]
Therefore, the completely factored form of the polynomial is
[tex]$$3(x-3)^2.$$[/tex]
The correct choice is:
A. [tex]$$3x^2-18x+27 = 3(x-3)^2.$$[/tex]
[tex]$$3x^2 - 18x + 27,$$[/tex]
we proceed as follows:
1. Factor out the greatest common factor (GCF):
All the coefficients (3, -18, 27) have a common factor of 3. Thus, we factor 3 out of the polynomial:
[tex]$$3x^2 - 18x + 27 = 3(x^2 - 6x + 9).$$[/tex]
2. Factor the quadratic inside the parentheses:
Next, we need to factor the quadratic expression
[tex]$$x^2 - 6x + 9.$$[/tex]
Look for two numbers that multiply to [tex]$9$[/tex] (the constant term) and add to [tex]$-6$[/tex] (the coefficient of [tex]$x$[/tex]). The numbers [tex]$-3$[/tex] and [tex]$-3$[/tex] satisfy these conditions since:
[tex]$$(-3) \times (-3) = 9$$[/tex]
and
[tex]$$(-3) + (-3) = -6.$$[/tex]
Thus, the quadratic factors nicely as:
[tex]$$x^2 - 6x + 9 = (x-3)(x-3) = (x-3)^2.$$[/tex]
3. Write the complete factorization:
Replace the factored quadratic back into the expression:
[tex]$$3x^2 - 18x + 27 = 3(x-3)^2.$$[/tex]
Therefore, the completely factored form of the polynomial is
[tex]$$3(x-3)^2.$$[/tex]
The correct choice is:
A. [tex]$$3x^2-18x+27 = 3(x-3)^2.$$[/tex]
