Answer :
We want to factor the cubic polynomial
[tex]$$
f(x)=6x^3+19x^2-19x+4
$$[/tex]
into linear factors.
Step 1. Find a Linear Factor
To begin, we look for a value [tex]$c$[/tex] such that [tex]$f(-c)=0$[/tex]. Testing [tex]$x=-4$[/tex], we compute:
[tex]$$
f(-4)=6(-4)^3+19(-4)^2-19(-4)+4.
$$[/tex]
Simplifying:
[tex]\[
\begin{aligned}
6(-4)^3 &= 6(-64) = -384,\\[1mm]
19(-4)^2 &= 19\cdot16 = 304,\\[1mm]
-19(-4) &= 76,\\[1mm]
\text{and }+4 &= 4.
\end{aligned}
\][/tex]
Thus,
[tex]$$
f(-4)=-384+304+76+4=0.
$$[/tex]
Since [tex]$f(-4)=0$[/tex], by the Factor Theorem it follows that [tex]$(x+4)$[/tex] is a factor.
Step 2. Divide by the Factor
Divide the polynomial [tex]$f(x)$[/tex] by [tex]$(x+4)$[/tex] to find the quadratic factor. This division gives:
[tex]$$
\frac{f(x)}{x+4}=6x^2-5x+1.
$$[/tex]
Step 3. Factor the Quadratic
We now factor the quadratic polynomial:
[tex]$$
6x^2-5x+1.
$$[/tex]
We look for two numbers that multiply to [tex]$6 \cdot 1 = 6$[/tex] and add to [tex]$-5$[/tex]. It turns out that the quadratic factors as:
[tex]$$
6x^2-5x+1=(2x-1)(3x-1).
$$[/tex]
Verifying by multiplication:
[tex]\[
(2x-1)(3x-1)=6x^2-2x-3x+1=6x^2-5x+1.
\][/tex]
Step 4. Write the Complete Factorization
Putting everything together, we have factored the polynomial as:
[tex]$$
f(x)=(x+4)(2x-1)(3x-1).
$$[/tex]
Thus, the complete factorization of the polynomial is:
[tex]$$
\boxed{(x+4)(2x-1)(3x-1)}.
$$[/tex]
[tex]$$
f(x)=6x^3+19x^2-19x+4
$$[/tex]
into linear factors.
Step 1. Find a Linear Factor
To begin, we look for a value [tex]$c$[/tex] such that [tex]$f(-c)=0$[/tex]. Testing [tex]$x=-4$[/tex], we compute:
[tex]$$
f(-4)=6(-4)^3+19(-4)^2-19(-4)+4.
$$[/tex]
Simplifying:
[tex]\[
\begin{aligned}
6(-4)^3 &= 6(-64) = -384,\\[1mm]
19(-4)^2 &= 19\cdot16 = 304,\\[1mm]
-19(-4) &= 76,\\[1mm]
\text{and }+4 &= 4.
\end{aligned}
\][/tex]
Thus,
[tex]$$
f(-4)=-384+304+76+4=0.
$$[/tex]
Since [tex]$f(-4)=0$[/tex], by the Factor Theorem it follows that [tex]$(x+4)$[/tex] is a factor.
Step 2. Divide by the Factor
Divide the polynomial [tex]$f(x)$[/tex] by [tex]$(x+4)$[/tex] to find the quadratic factor. This division gives:
[tex]$$
\frac{f(x)}{x+4}=6x^2-5x+1.
$$[/tex]
Step 3. Factor the Quadratic
We now factor the quadratic polynomial:
[tex]$$
6x^2-5x+1.
$$[/tex]
We look for two numbers that multiply to [tex]$6 \cdot 1 = 6$[/tex] and add to [tex]$-5$[/tex]. It turns out that the quadratic factors as:
[tex]$$
6x^2-5x+1=(2x-1)(3x-1).
$$[/tex]
Verifying by multiplication:
[tex]\[
(2x-1)(3x-1)=6x^2-2x-3x+1=6x^2-5x+1.
\][/tex]
Step 4. Write the Complete Factorization
Putting everything together, we have factored the polynomial as:
[tex]$$
f(x)=(x+4)(2x-1)(3x-1).
$$[/tex]
Thus, the complete factorization of the polynomial is:
[tex]$$
\boxed{(x+4)(2x-1)(3x-1)}.
$$[/tex]
