50,838 views
35 votes
35 votes
(x+2)/(x^3+5x^2-4x-15)

Final exams are a comin. I just need a step by step explanation for this.

User Radoslav Stoyanov
by
2.6k points

1 Answer

25 votes
25 votes

Answer:

Look at the step by step

Explanation:

To solve this fraction, we first need to find the value of x that makes the denominator equal to zero. This is because if the denominator is equal to zero, the fraction will be undefined.

To find the value of x that makes the denominator equal to zero, we need to set the denominator equal to zero and solve for x. This gives us the equation:

x^3 + 5x^2 - 4x - 15 = 0

To solve this equation, we can use a number of methods, such as factoring or using the quadratic formula. One way to solve this equation is by using the Rational Root Theorem, which states that if a rational number p/q is a root of a polynomial with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

In this case, the constant term is -15 and the leading coefficient is 1, so the possible rational roots are -1, -3, -5, -15, 1, 3, 5, and 15. We can try each of these values for x to see if they are roots of the polynomial.

If we try x = -1, we get the equation (-1)^3 + 5(-1)^2 - 4(-1) - 15 = 0, which simplifies to -1 + 5 + 4 + 15 = 0. This does not equal zero, so -1 is not a root of the polynomial.

If we try x = -3, we get the equation (-3)^3 + 5(-3)^2 - 4(-3) - 15 = 0, which simplifies to -27 + 45 + 12 + 15 = 0. This does not equal zero, so -3 is not a root of the polynomial.

If we try x = -5, we get the equation (-5)^3 + 5(-5)^2 - 4(-5) - 15 = 0, which simplifies to -125 + 125 - 20 + 15 = 0. This equals zero, so -5 is a root of the polynomial.

If we try x = -15, we get the equation (-15)^3 + 5(-15)^2 - 4(-15) - 15 = 0, which simplifies to -3375 + 225 - 60 + 15 = 0. This does not equal zero, so -15 is not a root of the polynomial.

If we try x = 1, we get the equation 1^3 + 5(1)^2 - 4(1) - 15 = 0, which simplifies to 1 + 5 - 4 - 15 = 0. This does not equal zero, so 1 is not a root of the polynomial.

If we try x = 3, we get the equation 3^3 + 5(3)^2 - 4(3) - 15 = 0, which simplifies to 27 + 45 - 12 - 15 = 0. This does not equal zero, so 3 is not a root of the polynomial.

If we try x = 5, we get the equation 5^3 + 5(5)^2 - 4(5) - 15 = 0, which simplifies to 125 + 125 - 20 - 15 = 0. This equals zero, so 5 is a root of the polynomial.

If we try x = 15, we get the equation 15^3 + 5(15)^2 - 4(15) - 15 = 0, which simplifies to 3375 + 225 - 60 - 15 = 0.

User Jorge Quintana
by
3.2k points