Practice Your Skill: Factor completely- 1 62 + 8b + 7 2 n^2 + 1ln + 10 3 n^2 – 10n + 9 4. n^2 + 4n – 12 5. m^2 + m – 9 6. m^2 + Zm – 24 7. x^2 – 4x + 24 8 k^2 – 13k + 40 9. a^2 + 1la + 18 10. n^2 – n – 56
The Correct Answer and Explanation is:
Here are the completely factored forms of the given expressions:
- b2+8b+7=(b+1)(b+7)b^2 + 8b + 7 = (b + 1)(b + 7)
- n2−11n+10=(n−1)(n−10)n^2 – 11n + 10 = (n – 1)(n – 10)
- n2−10n+9=(n−1)(n−9)n^2 – 10n + 9 = (n – 1)(n – 9)
- n2+4n−12=(n+6)(n−2)n^2 + 4n – 12 = (n + 6)(n – 2)
- m2+m−90=(m+10)(m−9)m^2 + m – 90 = (m + 10)(m – 9)
- m2+2m−24=(m+6)(m−4)m^2 + 2m – 24 = (m + 6)(m – 4)
- x2−4x+24x^2 – 4x + 24 cannot be factored over the integers
- k2−13k+40=(k−5)(k−8)k^2 – 13k + 40 = (k – 5)(k – 8)
- a2+11a+18=(a+2)(a+9)a^2 + 11a + 18 = (a + 2)(a + 9)
- n2−n−56=(n−8)(n+7)n^2 – n – 56 = (n – 8)(n + 7)
Explanation:
Factoring quadratic trinomials relies on identifying two binomials whose product yields the original expression. For a general quadratic of the form ax2+bx+cax^2 + bx + c, we look for two numbers that multiply to acac and add to bb. In these problems, all leading coefficients aa are 1, which simplifies the process.
Take example 4: n2+4n−12n^2 + 4n – 12. We search for two numbers that multiply to -12 and sum to +4. The pair 6 and -2 fits. Thus, we rewrite it as (n+6)(n−2)(n + 6)(n – 2).
In example 7, x2−4x+24x^2 – 4x + 24, no integer pair multiplies to 24 and adds to -4. Since its discriminant (−4)2−4(1)(24)=16−96=−80(-4)^2 – 4(1)(24) = 16 – 96 = -80 is negative, it has no real roots and cannot be factored over the integers.
This process reinforces algebraic fluency and pattern recognition. Factoring is essential for solving quadratic equations, simplifying expressions, and analyzing polynomial functions. Mastery of it provides a foundation for more advanced topics like completing the square, quadratic formula application, and graphing.
