F(x), has (x-7) and (x+4i) as factors. f(x)=2x^5-13x^4+22x^3-187x^2-160x+3 36 what is the total number of real numbers of real zeros of f(x)

The zeros of a polynomial function are the factors of the function.

The number of real zeros of f(x) is 3

The function is given as:

Two of the factors are given as:

x - 7 ad x + 4i

The factor x + 4i is a complex factor.

If a polynomial function has a complex factor, then the conjugate of the complex factor is also a factor of the polynomial.

This means that f(x) has 2 complex factors

The degree of the polynomial is 5.

So, the number (r) of real zeros is the difference between the degree of the polynomial and the number of complex factors.

This gives

Hence, the number of real zeros of f(x) is 3

Read more about zeros of polynomials at:

link

3

Step-by-step explanation:

(x + 4i) is a root which means its conjugate (x - 4i) is also a root

(x + 4i)(x - 4i) = x² + 16

Use Long Division for 2x⁵ - 13x⁴ + 22x³ - 187x² - 160x + 336 ÷ x² + 16

            2x³ - 13x² - 10x + 21                              

x² + 16 ) 2x⁵ - 13x⁴ + 22x³ - 187x² - 160x + 336

           - (2x⁵ + 0x⁴ + 32x³)   ↓          ↓         ↓

                       -13x⁴ - 10x³ - 187x²     ↓         ↓

                    - (-13x⁴ +0x³ - 208x²)    ↓         ↓

                                  -10x³ + 21x² - 160x     ↓

                               - (-10x³  + 0x² - 160x)    ↓

                                               21x² + 0x + 336

                                            - (21x² + 0x + 336)

                                                                      0

Use Synthetic Division to divide (x - 7) by the reduced polynomial (2x³ - 13x² - 10x + 21)

7 |  2   -13   -10   21

  |  ↓    14     7   -21        

    2      1     -3     0  ← remainder is 0      

Use grouping to factor the reduced polynomial (2x² + x - 3)  

 2x² - 2x + 3x - 3

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

= (2x + 1)(x - 1)  

The factors are: (x - 7) (2x + 1) (x - 1) (x + 4i) (x - 4i)

The first 3 are real roots and the last 2 are complex roots

step-by-step explanation:

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