Question: Written Exercises Factor. Check by multiplying the factors. If the polynomial is not…

Written Exercises

Factor. Check by multiplying the factors. If the polynomial is not factorable, write prime.

1. \(x^2 + 5x + 4\)
2. \(z^2 + 9z + 8\)
3. \(r^2 - 6r + 8\)
4. \(c^2 - 10c + 16\)
5. \(y^2 - 9y + 14\)
6. \(p^2 - 14p + 13\)
7. \(q^2 + 16q + 15\)
8. \(n^2 + 10n + 21\)
9. \(a^2 - 15a + 26\)
10. \(s^2 - 12s + 40\)
11. \(x^2 + 20x + 36\)
12. \(z^2 + 16z + 39\)
13. \(u^2 + 12u + 28\)
14. \(x^2 - 22x + 72\)
15. \(42 - 23k + k^2\)
16. \(64 - 20s + s^2\)
17. \(75 + 20r + r^2\)
18. \(75 + 27u + u^2\)

Solution

Let’s solve problem 1: Given the quadratic expression \(x^2 + 5x + 4\). Start by identifying numbers that multiply to 4 and add to 5. Those numbers are 1 and 4. Rewrite the middle term using these numbers: \(x^2 + 1x + 4x + 4\). Group the terms: \((x^2 + 1x) + (4x + 4)\). Factor out the common factors in each group: \(x(x + 1) + 4(x + 1)\). Factor out the common binomial factor: \((x + 1)(x + 4)\). Thus, the factorization of \(x^2 + 5x + 4\) is \((x + 1)(x + 4)\).