Factor x^2+2x+42

Problem

x2+2*x+42

Solution

1. Identify the coefficients of the quadratic expression in the form a*x2+b*x+c Here, a=1 b=2 and c=42

2. Search for two integers that multiply to c=42 and add to b=2

3. List the factor pairs of 42: (1,42) (2,21) (3,14) and (6,7)

4. Check the sums of these pairs: 1+42=43 2+21=23 3+14=17 and 6+7=13 None of these sums equal 2.

5. Evaluate the discriminant D=b2−4*a*c to check for real roots.

D=2−4*(1)*(42)

D=4−168

D=−164

6. Conclude that since the discriminant is negative and there are no integer factors that satisfy the conditions, the trinomial is irreducible over the set of integers.

Final Answer

x2+2*x+42=prime (irreducible over integers)

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