Here are the step-by-step solutions for factoring each trinomial.
1) $3p^2 - 2p - 5$
* Multiply the first coefficient (3) by the last number (-5): $3 \times -5 = -15$.
* Find two numbers that multiply to -15 and add up to the middle number (-2). These numbers are
-5 and
3.
* Rewrite the middle term using these numbers: $3p^2 - 5p + 3p - 5$.
* Factor by grouping:
* Group the first two terms: $p(3p - 5)$
* Group the last two terms: $+1(3p - 5)$
* Combine the outside parts: $(p + 1)(3p - 5)$.
2) $2n^2 + 3n - 9$
* Multiply $2 \times -9 = -18$.
* Find two numbers that multiply to -18 and add to 3. These are
6 and
-3.
* Rewrite the middle term: $2n^2 + 6n - 3n - 9$.
* Factor by grouping:
* $2n(n + 3)$
* $-3(n + 3)$
* Combine: $(2n - 3)(n + 3)$.
3) $3n^2 - 8n + 4$
* Multiply $3 \times 4 = 12$.
* Find two numbers that multiply to 12 and add to -8. Since the product is positive and sum is negative, both numbers must be negative. They are
-6 and
-2.
* Rewrite the middle term: $3n^2 - 6n - 2n + 4$.
* Factor by grouping:
* $3n(n - 2)$
* $-2(n - 2)$
* Combine: $(3n - 2)(n - 2)$.
4) $5n^2 + 19n + 12$
* Multiply $5 \times 12 = 60$.
* Find two numbers that multiply to 60 and add to 19. They are
15 and
4.
* Rewrite the middle term: $5n^2 + 15n + 4n + 12$.
* Factor by grouping:
* $5n(n + 3)$
* $+4(n + 3)$
* Combine: $(5n + 4)(n + 3)$.
5) $2v^2 + 11v + 5$
* Multiply $2 \times 5 = 10$.
* Find two numbers that multiply to 10 and add to 11. They are
10 and
1.
* Rewrite the middle term: $2v^2 + 10v + 1v + 5$.
* Factor by grouping:
* $2v(v + 5)$
* $+1(v + 5)$
* Combine: $(2v + 1)(v + 5)$.
6) $2n^2 + 5n + 2$
* Multiply $2 \times 2 = 4$.
* Find two numbers that multiply to 4 and add to 5. They are
4 and
1.
* Rewrite the middle term: $2n^2 + 4n + 1n + 2$.
* Factor by grouping:
* $2n(n + 2)$
* $+1(n + 2)$
* Combine: $(2n + 1)(n + 2)$.
7) $7a^2 + 53a + 28$
* Multiply $7 \times 28 = 196$.
* Find two numbers that multiply to 196 and add to 53.
* Factors of 196 include: 1 & 196, 2 & 98, 4 & 49...
* $4 + 49 = 53$. So the numbers are
49 and
4.
* Rewrite the middle term: $7a^2 + 49a + 4a + 28$.
* Factor by grouping:
* $7a(a + 7)$
* $+4(a + 7)$
* Combine: $(7a + 4)(a + 7)$.
8) $9k^2 + 66k + 21$
*
Check for a Greatest Common Factor (GCF) first. All three numbers (9, 66, 21) are divisible by 3.
* Factor out the 3: $3(3k^2 + 22k + 7)$.
* Now factor the part inside the parentheses: $3k^2 + 22k + 7$.
* Multiply $3 \times 7 = 21$.
* Find two numbers that multiply to 21 and add to 22. They are
21 and
1.
* Rewrite the middle term: $3k^2 + 21k + 1k + 7$.
* Factor by grouping:
* $3k(k + 7)$
* $+1(k + 7)$
* Inside factors: $(3k + 1)(k + 7)$.
* Don't forget the 3 we pulled out at the start.
* Final result: $3(3k + 1)(k + 7)$.
Final Answer:
1) $(3p - 5)(p + 1)$
2) $(2n - 3)(n + 3)$
3) $(3n - 2)(n - 2)$
4) $(5n + 4)(n + 3)$
5) $(2v + 1)(v + 5)$
6) $(2n + 1)(n + 2)$
7) $(7a + 4)(a + 7)$
8) $3(3k + 1)(k + 7)$
Parent Tip: Review the logic above to help your child master the concept of algebra 1 factoring trinomials worksheet.
