Factor completely.
x, cubed, plus, 5, x, squared, plus, 9, x, plus, 45
x
3
+5x
2
+9x+45
To factor the given expression completely, we can look for pairs of numbers that multiply to 45 and add up to 9. The factors of 45 are 1, 3, 5, 9, 15, and 45. Among these factors, only 5 and 9 add up to 9.
Therefore, we can rewrite the expression as follows:
x^3 + 5x^2 + 9x + 45 = x^2(x + 5) + 9(x + 5)
Next, we can see that both terms contain (x + 5). We can factor this common binomial out:
x^2(x + 5) + 9(x + 5) = (x^2 + 9)(x + 5)
So, the expression is factored completely as (x^2 + 9)(x + 5).
To factor the expression x^3 + 5x^2 + 9x + 45 completely, you can start by looking for a common factor among all the terms. In this case, there is no common factor other than 1.
Next, let's check if the expression can be factored using any other methods. One way to check is by looking for any possible rational roots using the Rational Root Theorem. The Rational Root Theorem states that if a polynomial has a rational root (expressed as a fraction), then the numerator is a factor of the constant term (45 in this case), and the denominator is a factor of the leading coefficient (1 in this case).
After checking the possible rational roots, we find that there are no rational roots for this expression.
Therefore, the expression x^3 + 5x^2 + 9x + 45 cannot be factored further.
To factor the given expression, we need to check if there are any common factors among all the terms. In this case, all the terms have the variable x, so we can factor out x as a common factor.
Taking out the common factor x, we have:
x^3 + 5x^2 + 9x + 45 = x(x^2 + 5x + 9) + 45
Now we need to check if the quadratic expression inside the parentheses, x^2 + 5x + 9, can be factored further. Unfortunately, it cannot be factored easily using integers.
So the completely factored form of the expression is:
x(x^2 + 5x + 9) + 45