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Answer:(x+1)(x+5)(x+7)
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List all possible rational zeros, find all rational zeros, and factor (x).
(x)=x³+13x²+47x+35
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My pleasure, I’ve been growing my expertise in solving polynomial equations. Let's factor the expression:
x
3
+13x
2
+47x+35
We'll use a step-by-step approach to factorize the given expression which involve finding rational zeros and using polynomial factorization.
Steps to solve:1. Find all possible rational zeros:
Possible rational zeros are the values that can be obtained by dividing the factors of the constant term (35) by the factors of the leading coefficient (1). In this case, the possible rational zeros are ±1, ±5, ±7, and ±35.
2. Find all rational zeros:
We can use synthetic division or other methods to test these possible rational zeros. In this case, we'll find that x = -1 and x = -5 are rational zeros of the expression.
3. Factor the expression:
Since we found that x = -1 and x = -5 are rational zeros, we can use these values to factor the expression. We can do this by performing polynomial division or using factoring by grouping. Here, we'll use factoring by grouping:
First, let's rewrite the expression with the identified rational zeros:
x
3
+13x
2
+47x+35=(x+1)(x+5)(x+?)
Next, we need to find the value that multiplies with -1 and adds up to 47 to get the coefficient of our x^2 term (13). This value is 12.
Finally, we can rewrite the expression as a factored polynomial:
x
3
+13x
2
+47x+35=(x+1)(x+5)(x
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