To find the factors of the polynomial q³ + q² + 2q + 2, we can start by attempting polynomial factorization. One effective method is to apply the technique of grouping.
1. **Grouping Terms**: Begin by grouping the terms in pairs:
(q³ + q²) + (2q + 2)2. **Factoring Out Common Factors**: Now, factor out the common factors from each group:
q²(q + 1) + 2(q + 1)3. **Factoring by Grouping**: Notice that both groups contain the common factor (q + 1). We can factor this out:
(q + 1)(q² + 2)Now, we have expressed the polynomial as the product of two factors: (q + 1) and (q² + 2).
4. **Final Check**: We can check our factorization by multiplying the factors back together. If we perform the multiplication:
(q + 1)(q² + 2) = q³ + q² + 2q + 2This confirms that the factorization is correct.
In conclusion, the factors of the polynomial q³ + q² + 2q + 2 are:
- q + 1
- q² + 2
This means that you can express the original polynomial as (q + 1)(q² + 2).
Additionally, the factor q² + 2 cannot be factored further using real numbers, as it has no real roots. Therefore, the fully factored form over the reals would be:
(q + 1)(q² + 2)