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Question

find the other zeros f(x)= x^3+5x^2-2x-10 
one zeros is -5 

List all of the possible rational zeros of the functions 
1.) f(x)=x^5-3x^2+1 
2.) f(x)=2x^4-3x^3-x+8

 

Answer 1

The function f(x) = x³ + 5x² - 2x - 10

- 5 is one zero.

Use synthetic division to find (x³ + 5x² - 2x - 10)/(x + 5)

Step 1: Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients as shown below.

Step 2: Write the constant r of the divisor (x - r) to the left. In this case, r = - 5. Bring the first coefficient, 1, down.

Step 3: Multiply the first coefficient by r : -5(1). Write the product under the second coefficient, 5 and : - 5 + 5 = 0.

Step 4: Multiply the sum,0, by r : (- 5)(0) = 0.

Write the product under the next coefficient, - 2 and add : - 2 + 0 = - 2.

Step 5: Multiply the sum, -2, by r : (- 5)(- 2) = 10.

Write the product under the next coefficient, 10 and add : . The remainder is 0.

Depressed polynomial is x²  - 2

Solve the equation x²  - 2 = 0

x²  =  2

Apply square root on each side.

x  = √ 2

x = ± 1.414

Other zeros of f(x) = x³ + 5x² - 2x - 10 are 1.414 and - 1.414.

Answer 2

1) The function f(x) = x⁵ - 3x² + 1

The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

If p/q is a rational zero, then p is a factor of 1 and q is a factor of 1.

The possible values of p are   ± 1.

The possible values for q are ± 1.

So, p/q = ± 1.

Possible rational zero is ± 1 .

Answer 3

2) The function f(x) = 2x⁴ - 3x³ - x + 8

The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

If p/q is a rational zero, then p is a factor of 8 and q is a factor of 2.

The possible values of p are  ± 1, ± 2, ± 4 and ± 8.

The possible values for q are ± 1 and ± 2.

So, p/q = ± 1, ± 2, ± 4 ,± 8 and ± 1/2.

Possible rational zeros are ± 1, ± 2, ± 4 ,± 8 and ± 1/2.