Question

1. $8n^3 + 28n^2 + 14n + 49$
2. $5x^3 - 20x^2 + 4x - 16$
3. $56x^3 - 40x^2 - 7x + 5$
4. $7n^3 - 49n^2 - 3n + 21$
5. $b^3 + 3b^2 + 4b + 12$
6. $7k^3 - 56k^2 - 3k + 24$

Explanation:

Let's solve each polynomial by factoring by grouping. We'll take problem 5 as an example first, then solve the others in order.

Problem 5: \(8n^3 + 28n^2 + 14n + 49\)

Step 1: Group terms

Group the first two terms and the last two terms: \((8n^3 + 28n^2) + (14n + 49)\)

Step 2: Factor out GCF from each group

Factor out \(4n^2\) from the first group and \(7\) from the second group: \(4n^2(2n + 7) + 7(2n + 7)\)

Step 3: Factor out the common binomial factor

Factor out \((2n + 7)\): \((4n^2 + 7)(2n + 7)\)

Step 1: Group terms

\((5x^3 - 20x^2) + (4x - 16)\)

Step 2: Factor out GCF from each group

Factor out \(5x^2\) from the first group and \(4\) from the second group: \(5x^2(x - 4) + 4(x - 4)\)

Step 3: Factor out the common binomial factor

Factor out \((x - 4)\): \((5x^2 + 4)(x - 4)\)

Step 1: Group terms

\((56x^3 - 40x^2) + (-7x + 5)\)

Step 2: Factor out GCF from each group

Factor out \(8x^2\) from the first group and \(-1\) from the second group: \(8x^2(7x - 5) - 1(7x - 5)\)

Step 3: Factor out the common binomial factor

Factor out \((7x - 5)\): \((8x^2 - 1)(7x - 5)\)

Answer:

\((4n^2 + 7)(2n + 7)\)

Problem 6: \(5x^3 - 20x^2 + 4x - 16\)