Let's solve each of these cubic equations by
factoring by grouping. This method involves grouping terms in pairs, factoring out the greatest common factor (GCF) from each pair, and then factoring out the common binomial.
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Problem 1: Factor
$$
x^3 + 3x^2 + x + 3
$$
Step 1: Group terms
Group the first two and last two terms:
$$
(x^3 + 3x^2) + (x + 3)
$$
Step 2: Factor out GCF from each group
$$
x^2(x + 3) + 1(x + 3)
$$
Step 3: Factor out the common binomial
$$
(x^2 + 1)(x + 3)
$$
✔ Answer: $$(x^2 + 1)(x + 3)$$
>
✔ Correct choice:
(x^2 + 1)(x + 3)
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Problem 2: Factor
$$
2x^3 + x^2 + 4x + 2
$$
Step 1: Group terms
$$
(2x^3 + x^2) + (4x + 2)
$$
Step 2: Factor out GCF from each group
$$
x^2(2x + 1) + 2(2x + 1)
$$
Step 3: Factor out the common binomial
$$
(x^2 + 2)(2x + 1)
$$
✔ Answer: $$(x^2 + 2)(2x + 1)$$
>
✔ Correct choice:
(x^2 + 2)(2x + 1)
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Problem 3: Factor
$$
x^3 - 4x^2 + 2x - 8
$$
Step 1: Group terms
$$
(x^3 - 4x^2) + (2x - 8)
$$
Step 2: Factor out GCF from each group
$$
x^2(x - 4) + 2(x - 4)
$$
Step 3: Factor out the common binomial
$$
(x^2 + 2)(x - 4)
$$
✔ Answer: $$(x^2 + 2)(x - 4)$$
>
✔ Correct choice:
(x^2 + 2)(x - 4)
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✔ Final Answers:
1. $$(x^2 + 1)(x + 3)$$
2. $$(x^2 + 2)(2x + 1)$$
3. $$(x^2 + 2)(x - 4)$$
These are the correct factorizations using
factoring by grouping.
Parent Tip: Review the logic above to help your child master the concept of factoring cubic polynomials worksheet.
