Math worksheet for factoring trinomials, featuring ten algebraic expressions to be factored.
Worksheet titled "Factoring Trinomials" with ten numbered problems involving algebraic expressions to be factored, including variables like p, b, x, q, m, and s.
WEBP
742×1050
15.2 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #663801
⭐
Show Answer Key & Explanations
Step-by-step solution for: Factoring Trinomials Worksheets - Math Monks
▼
Show Answer Key & Explanations
Step-by-step solution for: Factoring Trinomials Worksheets - Math Monks
Here are the step-by-step solutions for factoring each trinomial on the worksheet.
1) $2p^2 + 6p - 108$
* Step 1: Look for a Greatest Common Factor (GCF). All numbers are divisible by 2.
$$2(p^2 + 3p - 54)$$
* Step 2: Factor the inside part $(p^2 + 3p - 54)$. We need two numbers that multiply to $-54$ and add to $3$.
* Factors of 54: $1, 54$; $2, 27$; $3, 18$; $6, 9$.
* Since the product is negative, one number is positive and one is negative. Since the sum is positive ($3$), the larger number is positive.
* Check $9$ and $-6$: $9 \times -6 = -54$ and $9 + (-6) = 3$. This works.
* Step 3: Write the factors.
$$2(p + 9)(p - 6)$$
2) $7b^2 + 53b + 28$
* Step 1: Multiply the first coefficient ($7$) by the last number ($28$): $7 \times 28 = 196$.
* Step 2: Find two numbers that multiply to $196$ and add to the middle number ($53$).
* Let's check factors of 196. $4 \times 49 = 196$.
* Check sum: $4 + 49 = 53$. This works.
* Step 3: Rewrite the middle term using these numbers: $7b^2 + 49b + 4b + 28$.
* Step 4: Factor by grouping.
* Group 1: $7b(b + 7)$
* Group 2: $4(b + 7)$
* Step 5: Combine the outer parts.
$$(7b + 4)(b + 7)$$
3) $6x^2 + 7x - 49$
* Step 1: Multiply first and last: $6 \times -49 = -294$.
* Step 2: Find two numbers that multiply to $-294$ and add to $7$.
* Let's try factors near $\sqrt{294} \approx 17$.
* $14 \times 21 = 294$.
* We need a positive sum, so use $+21$ and $-14$.
* Check: $21 + (-14) = 7$. This works.
* Step 3: Rewrite middle term: $6x^2 - 14x + 21x - 49$.
* Step 4: Factor by grouping.
* Group 1: $2x(3x - 7)$
* Group 2: $7(3x - 7)$
* Step 5: Combine.
$$(2x + 7)(3x - 7)$$
4) $3p^2 - 3p - 36$
* Step 1: Look for a GCF. All numbers are divisible by 3.
$$3(p^2 - p - 12)$$
* Step 2: Factor $(p^2 - p - 12)$. Find two numbers that multiply to $-12$ and add to $-1$.
* Factors of 12: $3, 4$.
* Use $-4$ and $+3$: $-4 \times 3 = -12$ and $-4 + 3 = -1$.
* Step 3: Write the factors.
$$3(p - 4)(p + 3)$$
5) $5q^2 + 19q + 12$
* Step 1: Multiply first and last: $5 \times 12 = 60$.
* Step 2: Find two numbers that multiply to $60$ and add to $19$.
* Factors of 60: $4 \times 15 = 60$.
* Check sum: $4 + 15 = 19$. This works.
* Step 3: Rewrite middle term: $5q^2 + 15q + 4q + 12$.
* Step 4: Factor by grouping.
* Group 1: $5q(q + 3)$
* Group 2: $4(q + 3)$
* Step 5: Combine.
$$(5q + 4)(q + 3)$$
6) $x^2 + 12x - 13$
* Step 1: The first coefficient is 1, so we just look for two numbers that multiply to $-13$ and add to $12$.
* Step 2: 13 is a prime number, so the only factors are 1 and 13.
* Use $+13$ and $-1$: $13 \times -1 = -13$ and $13 + (-1) = 12$.
* Step 3: Write the factors.
$$(x + 13)(x - 1)$$
7) $3x^2 - 12x - 135$
* Step 1: Look for a GCF. All numbers are divisible by 3.
$$3(x^2 - 4x - 45)$$
* Step 2: Factor $(x^2 - 4x - 45)$. Find two numbers that multiply to $-45$ and add to $-4$.
* Factors of 45: $5, 9$.
* Use $-9$ and $+5$: $-9 \times 5 = -45$ and $-9 + 5 = -4$.
* Step 3: Write the factors.
$$3(x - 9)(x + 5)$$
8) $5m^2 - 11m - 12$
* Step 1: Multiply first and last: $5 \times -12 = -60$.
* Step 2: Find two numbers that multiply to $-60$ and add to $-11$.
* Factors of 60: $4, 15$.
* Use $-15$ and $+4$: $-15 \times 4 = -60$ and $-15 + 4 = -11$.
* Step 3: Rewrite middle term: $5m^2 - 15m + 4m - 12$.
* Step 4: Factor by grouping.
* Group 1: $5m(m - 3)$
* Group 2: $4(m - 3)$
* Step 5: Combine.
$$(5m + 4)(m - 3)$$
9) $56p^2 + 64p + 8$
* Step 1: Look for a GCF. All numbers are even, and actually all are divisible by 8.
$$8(7p^2 + 8p + 1)$$
* Step 2: Factor $(7p^2 + 8p + 1)$. Find two numbers that multiply to $7 \times 1 = 7$ and add to $8$.
* The numbers are $7$ and $1$.
* Step 3: Rewrite middle term: $7p^2 + 7p + 1p + 1$.
* Step 4: Factor by grouping.
* Group 1: $7p(p + 1)$
* Group 2: $1(p + 1)$
* Step 5: Combine with the GCF from Step 1.
$$8(7p + 1)(p + 1)$$
10) $25s^2 + 30s - 16$
* Step 1: Multiply first and last: $25 \times -16 = -400$.
* Step 2: Find two numbers that multiply to $-400$ and add to $30$.
* Think of factors of 400. $10 \times 40 = 400$. Difference is 30.
* Use $+40$ and $-10$: $40 \times -10 = -400$ and $40 + (-10) = 30$.
* Step 3: Rewrite middle term: $25s^2 - 10s + 40s - 16$.
* Step 4: Factor by grouping.
* Group 1: $5s(5s - 2)$
* Group 2: $8(5s - 2)$
* Step 5: Combine.
$$(5s + 8)(5s - 2)$$
Final Answer:
1) $2(p + 9)(p - 6)$
2) $(7b + 4)(b + 7)$
3) $(2x + 7)(3x - 7)$
4) $3(p - 4)(p + 3)$
5) $(5q + 4)(q + 3)$
6) $(x + 13)(x - 1)$
7) $3(x - 9)(x + 5)$
8) $(5m + 4)(m - 3)$
9) $8(7p + 1)(p + 1)$
10) $(5s + 8)(5s - 2)$
1) $2p^2 + 6p - 108$
* Step 1: Look for a Greatest Common Factor (GCF). All numbers are divisible by 2.
$$2(p^2 + 3p - 54)$$
* Step 2: Factor the inside part $(p^2 + 3p - 54)$. We need two numbers that multiply to $-54$ and add to $3$.
* Factors of 54: $1, 54$; $2, 27$; $3, 18$; $6, 9$.
* Since the product is negative, one number is positive and one is negative. Since the sum is positive ($3$), the larger number is positive.
* Check $9$ and $-6$: $9 \times -6 = -54$ and $9 + (-6) = 3$. This works.
* Step 3: Write the factors.
$$2(p + 9)(p - 6)$$
2) $7b^2 + 53b + 28$
* Step 1: Multiply the first coefficient ($7$) by the last number ($28$): $7 \times 28 = 196$.
* Step 2: Find two numbers that multiply to $196$ and add to the middle number ($53$).
* Let's check factors of 196. $4 \times 49 = 196$.
* Check sum: $4 + 49 = 53$. This works.
* Step 3: Rewrite the middle term using these numbers: $7b^2 + 49b + 4b + 28$.
* Step 4: Factor by grouping.
* Group 1: $7b(b + 7)$
* Group 2: $4(b + 7)$
* Step 5: Combine the outer parts.
$$(7b + 4)(b + 7)$$
3) $6x^2 + 7x - 49$
* Step 1: Multiply first and last: $6 \times -49 = -294$.
* Step 2: Find two numbers that multiply to $-294$ and add to $7$.
* Let's try factors near $\sqrt{294} \approx 17$.
* $14 \times 21 = 294$.
* We need a positive sum, so use $+21$ and $-14$.
* Check: $21 + (-14) = 7$. This works.
* Step 3: Rewrite middle term: $6x^2 - 14x + 21x - 49$.
* Step 4: Factor by grouping.
* Group 1: $2x(3x - 7)$
* Group 2: $7(3x - 7)$
* Step 5: Combine.
$$(2x + 7)(3x - 7)$$
4) $3p^2 - 3p - 36$
* Step 1: Look for a GCF. All numbers are divisible by 3.
$$3(p^2 - p - 12)$$
* Step 2: Factor $(p^2 - p - 12)$. Find two numbers that multiply to $-12$ and add to $-1$.
* Factors of 12: $3, 4$.
* Use $-4$ and $+3$: $-4 \times 3 = -12$ and $-4 + 3 = -1$.
* Step 3: Write the factors.
$$3(p - 4)(p + 3)$$
5) $5q^2 + 19q + 12$
* Step 1: Multiply first and last: $5 \times 12 = 60$.
* Step 2: Find two numbers that multiply to $60$ and add to $19$.
* Factors of 60: $4 \times 15 = 60$.
* Check sum: $4 + 15 = 19$. This works.
* Step 3: Rewrite middle term: $5q^2 + 15q + 4q + 12$.
* Step 4: Factor by grouping.
* Group 1: $5q(q + 3)$
* Group 2: $4(q + 3)$
* Step 5: Combine.
$$(5q + 4)(q + 3)$$
6) $x^2 + 12x - 13$
* Step 1: The first coefficient is 1, so we just look for two numbers that multiply to $-13$ and add to $12$.
* Step 2: 13 is a prime number, so the only factors are 1 and 13.
* Use $+13$ and $-1$: $13 \times -1 = -13$ and $13 + (-1) = 12$.
* Step 3: Write the factors.
$$(x + 13)(x - 1)$$
7) $3x^2 - 12x - 135$
* Step 1: Look for a GCF. All numbers are divisible by 3.
$$3(x^2 - 4x - 45)$$
* Step 2: Factor $(x^2 - 4x - 45)$. Find two numbers that multiply to $-45$ and add to $-4$.
* Factors of 45: $5, 9$.
* Use $-9$ and $+5$: $-9 \times 5 = -45$ and $-9 + 5 = -4$.
* Step 3: Write the factors.
$$3(x - 9)(x + 5)$$
8) $5m^2 - 11m - 12$
* Step 1: Multiply first and last: $5 \times -12 = -60$.
* Step 2: Find two numbers that multiply to $-60$ and add to $-11$.
* Factors of 60: $4, 15$.
* Use $-15$ and $+4$: $-15 \times 4 = -60$ and $-15 + 4 = -11$.
* Step 3: Rewrite middle term: $5m^2 - 15m + 4m - 12$.
* Step 4: Factor by grouping.
* Group 1: $5m(m - 3)$
* Group 2: $4(m - 3)$
* Step 5: Combine.
$$(5m + 4)(m - 3)$$
9) $56p^2 + 64p + 8$
* Step 1: Look for a GCF. All numbers are even, and actually all are divisible by 8.
$$8(7p^2 + 8p + 1)$$
* Step 2: Factor $(7p^2 + 8p + 1)$. Find two numbers that multiply to $7 \times 1 = 7$ and add to $8$.
* The numbers are $7$ and $1$.
* Step 3: Rewrite middle term: $7p^2 + 7p + 1p + 1$.
* Step 4: Factor by grouping.
* Group 1: $7p(p + 1)$
* Group 2: $1(p + 1)$
* Step 5: Combine with the GCF from Step 1.
$$8(7p + 1)(p + 1)$$
10) $25s^2 + 30s - 16$
* Step 1: Multiply first and last: $25 \times -16 = -400$.
* Step 2: Find two numbers that multiply to $-400$ and add to $30$.
* Think of factors of 400. $10 \times 40 = 400$. Difference is 30.
* Use $+40$ and $-10$: $40 \times -10 = -400$ and $40 + (-10) = 30$.
* Step 3: Rewrite middle term: $25s^2 - 10s + 40s - 16$.
* Step 4: Factor by grouping.
* Group 1: $5s(5s - 2)$
* Group 2: $8(5s - 2)$
* Step 5: Combine.
$$(5s + 8)(5s - 2)$$
Final Answer:
1) $2(p + 9)(p - 6)$
2) $(7b + 4)(b + 7)$
3) $(2x + 7)(3x - 7)$
4) $3(p - 4)(p + 3)$
5) $(5q + 4)(q + 3)$
6) $(x + 13)(x - 1)$
7) $3(x - 9)(x + 5)$
8) $(5m + 4)(m - 3)$
9) $8(7p + 1)(p + 1)$
10) $(5s + 8)(5s - 2)$
Parent Tip: Review the logic above to help your child master the concept of factoring quadratics worksheet pdf.
![Free Printable Factoring Quadratics Worksheets [PDFs] Brighterly](/images2/factoring-quadratics-worksheet-pdf/factoring-quadratics-worksheet-pdf-9.jpg)
