How do I factor to find the zeros

Question

y= 4x^3 + 4x^2 - 7x +2

find all the real zeros. 

I dont know how to factor this fraction. I saw the Ac factoring thing but I didnt understand it. May someone explain to me please?

Answer 1

y = 4x³ + 4x² - 7x + 2

 

According to the Rational Zero Theorem, if p/q is a rational zero of the polynomial, then p is a divisor of the constant term, 2, and q is a divisor of the leading coefficient, 4.

 

So, p = ±(1, 2) and q = ±(1, 2, 4)

 

The possibilities for p/q are 1, -1, 2, -2, 1/2, -1/2, 1/4, -1/4

 

Checking the various possibilities, we see that 1/2 is a zero of the polynomial.  So, by the Factor Theorem, the polynomial has x - 1/2 as a factor.  

 

Divide synthetically by x - 1/2:

 

1/2⌋     4      4      -7      2

       _______2____3___-2

           4      6      -4      0

 

So, 4x³ + 4x² - 7x + 2 = (x - 1/2)(4x² + 6x - 4)

 

                                = 2(x - 1/2)(2x² + 3x - 2)

 

                                = 2(x - 1/2)(2x - 1)(x + 2)

 

                                = 4(x - 1/2)(x - 1/2)(x + 2)

 

Setting each factor equal to zero, we find that x = 1/2 or -2

 

ANSWER:  The real zeros of the given polynomial are 1/2 (with                            multiplicity 2) and -2 (with multiplicity 1).