Math / Algebra

QuestionFactor the trinomial completely. $7 v^{2}+52 v+21$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $7 v^{2}+52 v+21=$ $\square$ B. The polynomial is prime.

Studdy Solution

STEP 1

1. We are given a quadratic trinomial of the form $ax^2 + bx + c$ .
2. The goal is to factor the trinomial completely, if possible.
3. If the trinomial cannot be factored using integers, it is considered prime.

STEP 2

1. Identify and apply the appropriate factoring method for the trinomial.
2. Check if the trinomial can be factored by grouping.
3. Verify the factorization or determine if the polynomial is prime.

STEP 3

Identify the trinomial: $7v^2 + 52v + 21$ .
Since the leading coefficient is not 1, we will use the method of factoring by grouping.

STEP 4

Multiply the leading coefficient (7) by the constant term (21):
$7 \times 21 = 147$
We need to find two numbers that multiply to 147 and add to the middle coefficient, 52.

STEP 5

Find two numbers that multiply to 147 and add to 52.
The numbers are 3 and 49 because:
$3 \times 49 = 147$ $3 + 49 = 52$
Rewrite the middle term, 52v, using these numbers:
$7v^2 + 3v + 49v + 21$

STEP 6

Factor by grouping:
Group the terms: $(7v^2 + 3v) + (49v + 21)$
Factor out the greatest common factor from each group:
$v(7v + 3) + 7(7v + 3)$
Notice that $(7v + 3)$ is a common factor.

STEP 7

Factor out the common factor:
$(v + 7)(7v + 3)$
This is the complete factorization of the trinomial.

STEP 8

Verify the factorization by expanding:
$(v + 7)(7v + 3) = v(7v + 3) + 7(7v + 3) = 7v^2 + 3v + 49v + 21 = 7v^2 + 52v + 21$
The factorization is verified.
The correct choice is: A. $7v^2 + 52v + 21 = (v + 7)(7v + 3)$

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QuestionFactor the trinomial completely. $7 v^{2}+52 v+21$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $7 v^{2}+52 v+21=$ $\square$ B. The polynomial is prime.

Studdy Solution

STEP 1

1. We are given a quadratic trinomial of the form $ax^2 + bx + c$ .
2. The goal is to factor the trinomial completely, if possible.
3. If the trinomial cannot be factored using integers, it is considered prime.

STEP 2

1. Identify and apply the appropriate factoring method for the trinomial.
2. Check if the trinomial can be factored by grouping.
3. Verify the factorization or determine if the polynomial is prime.

STEP 3

Identify the trinomial: $7v^2 + 52v + 21$ .
Since the leading coefficient is not 1, we will use the method of factoring by grouping.

STEP 4

Multiply the leading coefficient (7) by the constant term (21):
$7 \times 21 = 147$
We need to find two numbers that multiply to 147 and add to the middle coefficient, 52.

STEP 5

Find two numbers that multiply to 147 and add to 52.
The numbers are 3 and 49 because:
$3 \times 49 = 147$ $3 + 49 = 52$
Rewrite the middle term, 52v, using these numbers:
$7v^2 + 3v + 49v + 21$

STEP 6

Factor by grouping:
Group the terms: $(7v^2 + 3v) + (49v + 21)$
Factor out the greatest common factor from each group:
$v(7v + 3) + 7(7v + 3)$
Notice that $(7v + 3)$ is a common factor.

STEP 7

Factor out the common factor:
$(v + 7)(7v + 3)$
This is the complete factorization of the trinomial.

STEP 8

Verify the factorization by expanding:
$(v + 7)(7v + 3) = v(7v + 3) + 7(7v + 3) = 7v^2 + 3v + 49v + 21 = 7v^2 + 52v + 21$
The factorization is verified.
The correct choice is: A. $7v^2 + 52v + 21 = (v + 7)(7v + 3)$

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QuestionFactor the trinomial completely. 7v2+52v+217 v^{2}+52 v+217v2+52v+21Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 7v2+52v+21=7 v^{2}+52 v+21=7v2+52v+21= □\square□ B. The polynomial is prime.

Studdy Solution

STEP 1

1. We are given a quadratic trinomial of the form ax2+bx+c ax^2 + bx + c ax2+bx+c.2. The goal is to factor the trinomial completely, if possible.3. If the trinomial cannot be factored using integers, it is considered prime.

STEP 2

1. Identify and apply the appropriate factoring method for the trinomial.2. Check if the trinomial can be factored by grouping.3. Verify the factorization or determine if the polynomial is prime.

STEP 3

Identify the trinomial: 7v2+52v+21 7v^2 + 52v + 21 7v2+52v+21.Since the leading coefficient is not 1, we will use the method of factoring by grouping.

STEP 4

Multiply the leading coefficient (7) by the constant term (21):7×21=147 7 \times 21 = 147 7×21=147We need to find two numbers that multiply to 147 and add to the middle coefficient, 52.

STEP 5

Find two numbers that multiply to 147 and add to 52.The numbers are 3 and 49 because:3×49=147 3 \times 49 = 147 3×49=147 3+49=52 3 + 49 = 52 3+49=52Rewrite the middle term, 52v, using these numbers:7v2+3v+49v+21 7v^2 + 3v + 49v + 21 7v2+3v+49v+21

STEP 6

Factor by grouping:Group the terms: (7v2+3v)+(49v+21) (7v^2 + 3v) + (49v + 21) (7v2+3v)+(49v+21)Factor out the greatest common factor from each group:v(7v+3)+7(7v+3) v(7v + 3) + 7(7v + 3) v(7v+3)+7(7v+3)Notice that (7v+3) (7v + 3) (7v+3) is a common factor.

STEP 7

Factor out the common factor:(v+7)(7v+3) (v + 7)(7v + 3) (v+7)(7v+3)This is the complete factorization of the trinomial.

STEP 8

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QuestionFactor the trinomial completely. $7 v^{2}+52 v+21$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $7 v^{2}+52 v+21=$ $\square$ B. The polynomial is prime.

1. We are given a quadratic trinomial of the form $ax^2 + bx + c$ .
2. The goal is to factor the trinomial completely, if possible.
3. If the trinomial cannot be factored using integers, it is considered prime.

1. Identify and apply the appropriate factoring method for the trinomial.
2. Check if the trinomial can be factored by grouping.
3. Verify the factorization or determine if the polynomial is prime.

Identify the trinomial: $7v^2 + 52v + 21$ .
Since the leading coefficient is not 1, we will use the method of factoring by grouping.

Multiply the leading coefficient (7) by the constant term (21):
$7 \times 21 = 147$
We need to find two numbers that multiply to 147 and add to the middle coefficient, 52.

Find two numbers that multiply to 147 and add to 52.
The numbers are 3 and 49 because:
$3 \times 49 = 147$ $3 + 49 = 52$
Rewrite the middle term, 52v, using these numbers:
$7v^2 + 3v + 49v + 21$

Factor by grouping:
Group the terms: $(7v^2 + 3v) + (49v + 21)$
Factor out the greatest common factor from each group:
$v(7v + 3) + 7(7v + 3)$
Notice that $(7v + 3)$ is a common factor.

Factor out the common factor:
$(v + 7)(7v + 3)$
This is the complete factorization of the trinomial.