We appreciate your visit to Factorize tex 6x 3 25x 2 32x 12 tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
We want to factorize the polynomial
[tex]$$
6x^3 - 25x^2 + 32x - 12.
$$[/tex]
Step 1. Find a root using the Rational Root Theorem.
The Rational Root Theorem tells us that any rational root of the polynomial is of the form
[tex]$$
x = \frac{p}{q},
$$[/tex]
where the numerator [tex]$p$[/tex] is a factor of the constant term ([tex]$-12$[/tex]) and the denominator [tex]$q$[/tex] is a factor of the leading coefficient ([tex]$6$[/tex]). Testing some candidate values, we find that
[tex]$$
x = 2
$$[/tex]
is a root because
[tex]$$
6(2)^3 - 25(2)^2 + 32(2) - 12 = 48 - 100 + 64 - 12 = 0.
$$[/tex]
This tells us that [tex]$(x-2)$[/tex] is a factor of the polynomial.
Step 2. Divide the polynomial by [tex]$(x-2)$[/tex].
Dividing [tex]$$6x^3 - 25x^2 + 32x - 12$$[/tex] by [tex]$(x-2)$[/tex] (by synthetic division or polynomial long division) reduces the polynomial to a quadratic. After performing the division, we get:
[tex]$$
6x^3 - 25x^2 + 32x - 12 = (x-2)(6x^2 - 13x + 6).
$$[/tex]
Step 3. Factorize the quadratic [tex]$6x^2 - 13x + 6$[/tex].
To factor this quadratic, we look for two numbers that multiply to [tex]$$6 \cdot 6 = 36$$[/tex] and add up to [tex]$$-13.$$[/tex] The two numbers that work are [tex]$$-9$$[/tex] and [tex]$$-4$$[/tex] since
[tex]$$
-9 \times -4 = 36 \quad \text{and} \quad -9 + (-4) = -13.
$$[/tex]
Rewrite the middle term using these numbers:
[tex]$$
6x^2 - 13x + 6 = 6x^2 - 9x - 4x + 6.
$$[/tex]
Next, factor by grouping:
1. Group the first two and the last two terms:
[tex]$$
(6x^2 - 9x) + (-4x + 6).
$$[/tex]
2. Factor out the common factors in each group:
[tex]$$
3x(2x - 3) - 2(2x - 3).
$$[/tex]
3. Notice the common factor [tex]$(2x - 3)$[/tex]:
[tex]$$
(2x - 3)(3x - 2).
$$[/tex]
Thus, the quadratic factors as
[tex]$$
6x^2 - 13x + 6 = (2x-3)(3x-2).
$$[/tex]
Step 4. Write the complete factorization.
Substituting back, the complete factorization of the original polynomial is:
[tex]$$
6x^3 - 25x^2 + 32x - 12 = (x-2)(2x-3)(3x-2).
$$[/tex]
This is the factorized form of the polynomial.
[tex]$$
6x^3 - 25x^2 + 32x - 12.
$$[/tex]
Step 1. Find a root using the Rational Root Theorem.
The Rational Root Theorem tells us that any rational root of the polynomial is of the form
[tex]$$
x = \frac{p}{q},
$$[/tex]
where the numerator [tex]$p$[/tex] is a factor of the constant term ([tex]$-12$[/tex]) and the denominator [tex]$q$[/tex] is a factor of the leading coefficient ([tex]$6$[/tex]). Testing some candidate values, we find that
[tex]$$
x = 2
$$[/tex]
is a root because
[tex]$$
6(2)^3 - 25(2)^2 + 32(2) - 12 = 48 - 100 + 64 - 12 = 0.
$$[/tex]
This tells us that [tex]$(x-2)$[/tex] is a factor of the polynomial.
Step 2. Divide the polynomial by [tex]$(x-2)$[/tex].
Dividing [tex]$$6x^3 - 25x^2 + 32x - 12$$[/tex] by [tex]$(x-2)$[/tex] (by synthetic division or polynomial long division) reduces the polynomial to a quadratic. After performing the division, we get:
[tex]$$
6x^3 - 25x^2 + 32x - 12 = (x-2)(6x^2 - 13x + 6).
$$[/tex]
Step 3. Factorize the quadratic [tex]$6x^2 - 13x + 6$[/tex].
To factor this quadratic, we look for two numbers that multiply to [tex]$$6 \cdot 6 = 36$$[/tex] and add up to [tex]$$-13.$$[/tex] The two numbers that work are [tex]$$-9$$[/tex] and [tex]$$-4$$[/tex] since
[tex]$$
-9 \times -4 = 36 \quad \text{and} \quad -9 + (-4) = -13.
$$[/tex]
Rewrite the middle term using these numbers:
[tex]$$
6x^2 - 13x + 6 = 6x^2 - 9x - 4x + 6.
$$[/tex]
Next, factor by grouping:
1. Group the first two and the last two terms:
[tex]$$
(6x^2 - 9x) + (-4x + 6).
$$[/tex]
2. Factor out the common factors in each group:
[tex]$$
3x(2x - 3) - 2(2x - 3).
$$[/tex]
3. Notice the common factor [tex]$(2x - 3)$[/tex]:
[tex]$$
(2x - 3)(3x - 2).
$$[/tex]
Thus, the quadratic factors as
[tex]$$
6x^2 - 13x + 6 = (2x-3)(3x-2).
$$[/tex]
Step 4. Write the complete factorization.
Substituting back, the complete factorization of the original polynomial is:
[tex]$$
6x^3 - 25x^2 + 32x - 12 = (x-2)(2x-3)(3x-2).
$$[/tex]
This is the factorized form of the polynomial.
Thanks for taking the time to read Factorize tex 6x 3 25x 2 32x 12 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
