Factoring Trinomials Worksheets - Math Monks - Free Printable
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Step-by-step solution for: Factoring Trinomials Worksheets - Math Monks
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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.
1) $2p^2 + 6p - 108$
* Step 1: Look for a Greatest Common Factor (GCF). All numbers (2, 6, 108) are even, so we can divide 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 they multiply to a negative, one is positive and one is negative. Since they add to a positive (+3), the larger number is positive.
* Check: $-6 \times 9 = -54$ and $-6 + 9 = 3$. This works.
* Step 3: Write the factors.
$$2(p - 6)(p + 9)$$
2) $7b^2 + 53b + 28$
* Step 1: There is no GCF. We use the "ac method" or trial and error. Multiply $a \cdot c$: $7 \cdot 28 = 196$.
* Step 2: Find two numbers that multiply to 196 and add to 53.
* Let's check factors of 196: $4 \times 49 = 196$.
* Check sum: $4 + 49 = 53$. This works.
* Step 3: Rewrite the middle term ($53b$) using 4 and 49.
$$7b^2 + 49b + 4b + 28$$
* Step 4: Factor by grouping.
Group first two: $7b(b + 7)$
Group last two: $4(b + 7)$
* Step 5: Combine.
$$(7b + 4)(b + 7)$$
3) $6x^2 + 7x - 49$
* Step 1: Multiply $a \cdot c$: $6 \cdot -49 = -294$.
* Step 2: Find two numbers that multiply to -294 and add to 7.
* Let's try factors near $\sqrt{294}$ (approx 17).
* $14 \times 21 = 294$.
* To get +7, we need $+21$ and $-14$. ($21 - 14 = 7$).
* Step 3: Rewrite the middle term.
$$6x^2 - 14x + 21x - 49$$
* Step 4: Factor by grouping.
Group first two: $2x(3x - 7)$
Group last two: $7(3x - 7)$
* 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.
* To get -1, we need $-4$ and $+3$. ($-4 + 3 = -1$).
* Step 3: Write the factors.
$$3(p - 4)(p + 3)$$
5) $5q^2 + 19q + 12$
* Step 1: Multiply $a \cdot c$: $5 \cdot 12 = 60$.
* 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 the middle term.
$$5q^2 + 15q + 4q + 12$$
* Step 4: Factor by grouping.
Group first two: $5q(q + 3)$
Group last two: $4(q + 3)$
* Step 5: Combine.
$$(5q + 4)(q + 3)$$
6) $x^2 + 12x - 13$
* Step 1: The leading coefficient is 1, so we just look for two numbers that multiply to -13 and add to 12.
* 13 is prime, so the only factors are 1 and 13.
* To get +12, we need $+13$ and $-1$.
* Step 2: Write the factors directly.
$$(x + 13)(x - 1)$$
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.
* To get -4, we need $-9$ and $+5$. ($-9 + 5 = -4$).
* Step 3: Write the factors.
$$3(x - 9)(x + 5)$$
8) $5m^2 - 11m - 12$
* Step 1: Multiply $a \cdot c$: $5 \cdot -12 = -60$.
* Step 2: Find two numbers that multiply to -60 and add to -11.
* Factors of 60: ... $4 \times 15 = 60$.
* To get -11, we need $-15$ and $+4$. ($-15 + 4 = -11$).
* Step 3: Rewrite the middle term.
$$5m^2 - 15m + 4m - 12$$
* Step 4: Factor by grouping.
Group first two: $5m(m - 3)$
Group last two: $4(m - 3)$
* Step 5: Combine.
$$(5m + 4)(m - 3)$$
9) $56p^2 + 64p + 8$
* Step 1: Look for a GCF. All numbers are divisible by 8.
$$8(7p^2 + 8p + 1)$$
* Step 2: Factor $(7p^2 + 8p + 1)$. Find two numbers that multiply to $7 \cdot 1 = 7$ and add to 8.
* The numbers are 7 and 1.
* Step 3: Rewrite the middle term.
$$8(7p^2 + 7p + 1p + 1)$$
* Step 4: Factor by grouping inside the parenthesis.
Group first two: $7p(p + 1)$
Group last two: $1(p + 1)$
Inside becomes: $(7p + 1)(p + 1)$
* Step 5: Don't forget the 8 from Step 1.
$$8(7p + 1)(p + 1)$$
10) $25s^2 + 30s - 16$
* Step 1: Multiply $a \cdot c$: $25 \cdot -16 = -400$.
* 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.
* To get +30, we need $+40$ and $-10$.
* Step 3: Rewrite the middle term.
$$25s^2 - 10s + 40s - 16$$
* Step 4: Factor by grouping.
Group first two: $5s(5s - 2)$
Group last two: $8(5s - 2)$
* Step 5: Combine.
$$(5s + 8)(5s - 2)$$
──────────────────────────────────────
Final Answer:
1) $2(p - 6)(p + 9)$
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 (2, 6, 108) are even, so we can divide 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 they multiply to a negative, one is positive and one is negative. Since they add to a positive (+3), the larger number is positive.
* Check: $-6 \times 9 = -54$ and $-6 + 9 = 3$. This works.
* Step 3: Write the factors.
$$2(p - 6)(p + 9)$$
2) $7b^2 + 53b + 28$
* Step 1: There is no GCF. We use the "ac method" or trial and error. Multiply $a \cdot c$: $7 \cdot 28 = 196$.
* Step 2: Find two numbers that multiply to 196 and add to 53.
* Let's check factors of 196: $4 \times 49 = 196$.
* Check sum: $4 + 49 = 53$. This works.
* Step 3: Rewrite the middle term ($53b$) using 4 and 49.
$$7b^2 + 49b + 4b + 28$$
* Step 4: Factor by grouping.
Group first two: $7b(b + 7)$
Group last two: $4(b + 7)$
* Step 5: Combine.
$$(7b + 4)(b + 7)$$
3) $6x^2 + 7x - 49$
* Step 1: Multiply $a \cdot c$: $6 \cdot -49 = -294$.
* Step 2: Find two numbers that multiply to -294 and add to 7.
* Let's try factors near $\sqrt{294}$ (approx 17).
* $14 \times 21 = 294$.
* To get +7, we need $+21$ and $-14$. ($21 - 14 = 7$).
* Step 3: Rewrite the middle term.
$$6x^2 - 14x + 21x - 49$$
* Step 4: Factor by grouping.
Group first two: $2x(3x - 7)$
Group last two: $7(3x - 7)$
* 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.
* To get -1, we need $-4$ and $+3$. ($-4 + 3 = -1$).
* Step 3: Write the factors.
$$3(p - 4)(p + 3)$$
5) $5q^2 + 19q + 12$
* Step 1: Multiply $a \cdot c$: $5 \cdot 12 = 60$.
* 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 the middle term.
$$5q^2 + 15q + 4q + 12$$
* Step 4: Factor by grouping.
Group first two: $5q(q + 3)$
Group last two: $4(q + 3)$
* Step 5: Combine.
$$(5q + 4)(q + 3)$$
6) $x^2 + 12x - 13$
* Step 1: The leading coefficient is 1, so we just look for two numbers that multiply to -13 and add to 12.
* 13 is prime, so the only factors are 1 and 13.
* To get +12, we need $+13$ and $-1$.
* Step 2: Write the factors directly.
$$(x + 13)(x - 1)$$
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.
* To get -4, we need $-9$ and $+5$. ($-9 + 5 = -4$).
* Step 3: Write the factors.
$$3(x - 9)(x + 5)$$
8) $5m^2 - 11m - 12$
* Step 1: Multiply $a \cdot c$: $5 \cdot -12 = -60$.
* Step 2: Find two numbers that multiply to -60 and add to -11.
* Factors of 60: ... $4 \times 15 = 60$.
* To get -11, we need $-15$ and $+4$. ($-15 + 4 = -11$).
* Step 3: Rewrite the middle term.
$$5m^2 - 15m + 4m - 12$$
* Step 4: Factor by grouping.
Group first two: $5m(m - 3)$
Group last two: $4(m - 3)$
* Step 5: Combine.
$$(5m + 4)(m - 3)$$
9) $56p^2 + 64p + 8$
* Step 1: Look for a GCF. All numbers are divisible by 8.
$$8(7p^2 + 8p + 1)$$
* Step 2: Factor $(7p^2 + 8p + 1)$. Find two numbers that multiply to $7 \cdot 1 = 7$ and add to 8.
* The numbers are 7 and 1.
* Step 3: Rewrite the middle term.
$$8(7p^2 + 7p + 1p + 1)$$
* Step 4: Factor by grouping inside the parenthesis.
Group first two: $7p(p + 1)$
Group last two: $1(p + 1)$
Inside becomes: $(7p + 1)(p + 1)$
* Step 5: Don't forget the 8 from Step 1.
$$8(7p + 1)(p + 1)$$
10) $25s^2 + 30s - 16$
* Step 1: Multiply $a \cdot c$: $25 \cdot -16 = -400$.
* 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.
* To get +30, we need $+40$ and $-10$.
* Step 3: Rewrite the middle term.
$$25s^2 - 10s + 40s - 16$$
* Step 4: Factor by grouping.
Group first two: $5s(5s - 2)$
Group last two: $8(5s - 2)$
* Step 5: Combine.
$$(5s + 8)(5s - 2)$$
──────────────────────────────────────
Final Answer:
1) $2(p - 6)(p + 9)$
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 perfect square trinomials worksheet.
