Find all real zeros of the polynomial f(x) = x^4 + 12x^3 + 27x^2 and determine t

Question

Find all real zeros of the polynomial f(x) = x^4 + 12x^3 + 27x^2 and determine the multiplicity of each. Use long division to divide. (x^4 +7x^3 + 10x^2 - x - 2) + (x + 2) Determine whether the statement is true or false. Justify your answer. The rational expression x^4 + 2x^2 - 12x + 12/x^3 - 5x - 15 is improper. True. The degree of the numerator is greater than the degree of the denominator. False. The degree of the denominator is smaller than the degree of the numerator. If x = Squareroot 6 is a root of x^3 + 5x^2 - 6x - 30 = 0, use synthetic division to factor the polynomial list all real solutions of the equation. Find real numbers a and b such that the equation is true. (a - 4) + (b + 5)f = 13 + 11f

Explanation / Answer

(According to chegg policy, only four subquestions will be answered. Please post the remaining in another question)

1. x4 + 12x3 + 27x2 = 0

=> x2 (x2 + 12x + 27) = 0

=> x2 (x2 + 9x + 3x + 27) = 0

=>  x2 (x(x + 9) + 3(x + 9)) = 0

=> x2 (x+3) (x+9) = 0

Thus the zeros are 0,-3 and -9 with multiplicity 2,1 and 1 respectively.

2. x4 + 7x3 + 10x2 - x - 2

= x4 + 2x3 + 5x3 + 10x2 - x - 2

= x3 (x +2) + 5x (x+ 2) - 1 (x+2)

= (x3 + 5x - 1) (x + 2)

3. a. True. The degree of the numerator is greater than the degree of the denominator.

4. Since x = 6 is a root, we need to divide x3 + 5x2 -6x - 30 = 0 by x2 - 6

=> (x2 - 6) | (x3 + 5x2 -6x - 30) | x + 5

x3 -6x

-------------------------------------------------

5x2 - 30

5x2   - 30

---------------------------------------------------

0

Thus the quotient is x + 5

=> x3 + 5x2 -6x - 30 = (x2 - 6) (x + 5)

  

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.