Get detailed and reliable answers to your questions on IDNLearn.com. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.
Sagot :
The factorization of given cubic polynomial 6x³ - 11x² - 12x + 5 is:
6x³ - 11x² - 12x + 5 = (x + 1)(x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])
For given question,
We have been given the cubic polynomial 6x³ - 11x² - 12x + 5
We need to factorize given cubic polynomial.
By the rational roots theorem, any rational zero of f(x) is expressible in the form ± [tex]\frac{p}{q}[/tex] for integers p, q with p a divisor of the constant term 5 and q a divisor of the coefficient 6 of the leading term.
Factors of p = 5: 1, 5
Factors of q = 6: 1, 2, 3, 6
That means that the only possible rational zeros are:
±{ [tex]\frac{1}{1} ,\frac{1}{2} ,\frac{1}{3} ,\frac{1}{6} ,\frac{5}{1} ,\frac{5}{2} ,\frac{5}{3} ,\frac{5}{6}[/tex] }
= ±{ [tex]1 ,\frac{1}{2} ,\frac{1}{3} ,\frac{1}{6} ,5 ,\frac{5}{2} ,\frac{5}{3} ,\frac{5}{6}[/tex] }
We need to find the exact zeros of given cubic polynomial.
For x = 1,
6(1)³ - 11(1)² - 12(1) + 5 = -12
This means, x = 1 is not a zero of given cubic polynomial.
For x = -1,
6(-1)³ - 11(-1)² - 12(-1) + 5 = 0
This means, x = -1 is a zero of given cubic polynomial and (x + 1) is a factor.
To factorize given cubic polynomial we use synthetic division.
The synthetic division (6x³ - 11x² - 12x + 5) ÷ (x + 1) is as shown in following image.
⇒ 6x³ - 11x² - 12x + 5 = (x + 1)(6x² - 17x + 5)
The factors of above quadratic polynomial 6x² - 17x + 5 are:
⇒ 6x² - 17x + 5 = (x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])
So, the factors of given cubic polynomial are:
⇒ 6x³ - 11x² - 12x + 5 = (x + 1)(x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])
Therefore, the factorization of given cubic polynomial 6x³ - 11x² - 12x + 5 is: 6x³ - 11x² - 12x + 5 = (x + 1)(x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])
Learn more about the polynomial here:
https://brainly.com/question/16594008
#SPJ4

Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.