Question

kuta software - infinite algebra 1
factoring trinomials (a > 1)
factor each completely.

1. $3p^2 - 2p - 5$
2. $2n^2 + 3n - 9$
3. $3n^2 - 8n + 4$
4. $5n^2 + 19n + 12$
5. $2v^2 + 11v + 5$
6. $2n^2 + 5n + 2$
7. $7a^2 + 53a + 28$
8. $9k^2 + 66k + 21$

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Explanation:

Problem 2: $2n^2 + 3n - 9$

Step1: Find $a \cdot c$ and $b$

$a=2,\ b=3,\ c=-9$
$a \cdot c = 2 \times (-9) = -18$
Find two numbers that multiply to $-18$ and add to $3$: $6$ and $-3$

Step2: Split the middle term

$2n^2 + 6n - 3n - 9$

Step3: Group and factor

$2n(n + 3) - 3(n + 3)$

Step4: Factor out common binomial

$(n + 3)(2n - 3)$

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Problem 3: $3n^2 - 8n + 4$

Step1: Find $a \cdot c$ and $b$

$a=3,\ b=-8,\ c=4$
$a \cdot c = 3 \times 4 = 12$
Find two numbers that multiply to $12$ and add to $-8$: $-2$ and $-6$

Step2: Split the middle term

$3n^2 - 6n - 2n + 4$

Step3: Group and factor

$3n(n - 2) - 2(n - 2)$

Step4: Factor out common binomial

$(n - 2)(3n - 2)$

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Problem 4: $5n^2 + 19n + 12$

Step1: Find $a \cdot c$ and $b$

$a=5,\ b=19,\ c=12$
$a \cdot c = 5 \times 12 = 60$
Find two numbers that multiply to $60$ and add to $19$: $15$ and $4$

Step2: Split the middle term

$5n^2 + 15n + 4n + 12$

Step3: Group and factor

$5n(n + 3) + 4(n + 3)$

Step4: Factor out common binomial

$(n + 3)(5n + 4)$

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Problem 5: $2v^2 + 11v + 5$

Step1: Find $a \cdot c$ and $b$

$a=2,\ b=11,\ c=5$
$a \cdot c = 2 \times 5 = 10$
Find two numbers that multiply to $10$ and add to $11$: $10$ and $1$

Step2: Split the middle term

$2v^2 + 10v + v + 5$

Step3: Group and factor

$2v(v + 5) + 1(v + 5)$

Step4: Factor out common binomial

$(v + 5)(2v + 1)$

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Problem 6: $2n^2 + 5n + 2$

Step1: Find $a \cdot c$ and $b$

$a=2,\ b=5,\ c=2$
$a \cdot c = 2 \times 2 = 4$
Find two numbers that multiply to $4$ and add to $5$: $4$ and $1$

Step2: Split the middle term

$2n^2 + 4n + n + 2$

Step3: Group and factor

$2n(n + 2) + 1(n + 2)$

Step4: Factor out common binomial

$(n + 2)(2n + 1)$

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Problem 7: $7a^2 + 53a + 28$

Step1: Find $a \cdot c$ and $b$

$a=7,\ b=53,\ c=28$
$a \cdot c = 7 \times 28 = 196$
Find two numbers that multiply to $196$ and add to $53$: $49$ and $4$

Step2: Split the middle term

$7a^2 + 49a + 4a + 28$

Step3: Group and factor

$7a(a + 7) + 4(a + 7)$

Step4: Factor out common binomial

$(a + 7)(7a + 4)$

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Problem 8: $9k^2 + 66k + 21$

Step1: Factor out GCF first

GCF of $9, 66, 21$ is $3$
$3(3k^2 + 22k + 7)$

Step2: Factor the trinomial inside

For $3k^2 + 22k + 7$, $a \cdot c = 3 \times 7 = 21$
Find two numbers that multiply to $21$ and add to $22$: $21$ and $1$
Split middle term: $3k^2 + 21k + k + 7$
Group and factor: $3k(k + 7) + 1(k + 7) = (k + 7)(3k + 1)$

Step3: Combine with GCF

$3(k + 7)(3k + 1)$

Answer:

1. $(p+1)(3p-5)$
2. $(n + 3)(2n - 3)$
3. $(n - 2)(3n - 2)$
4. $(n + 3)(5n + 4)$
5. $(v + 5)(2v + 1)$
6. $(n + 2)(2n + 1)$
7. $(a + 7)(7a + 4)$
8. $3(k + 7)(3k + 1)$