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Step 1:I'll solve this step by step using the guidelines you specified:
Step 2:: To find which binomial is a factor, we'll use the factor theorem.
This means we'll test potential factors by checking which values of x make the polynomial equal to zero.
Step 3:: The potential factors of the constant term (- 15) are: ±1, ±3, ±5, ±15
Step 4:: We'll systematically test these potential factors by substituting them into the polynomial x^{2} + 2x - 15
Step 5:: Let's check x = 3:
3^{2} + 2(3) - 15 = 9 + 6 - 15 = 0
Step 6:: Since 3^{2} + 2(3) - 15 = 0, (x - 3) is a factor of the polynomial.
Step 7:: We can verify by polynomial long division or by factoring:
x^{2} + 2x - 15 = (x - 3)(x + 5)
Final Answer
The binomial (x - 3) is a factor of x^{2} + 2x - 15.
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