Math / Algebra

QuestionStep 1 need to divide. However, we note that the denominator can be factored as follows. $\begin{aligned} Q(x) & =6 x^{2}+7 x+1 \\ & =(6 x+1)(\square) \end{aligned}$ Submit Skip (you cannot come back)

Studdy Solution

STEP 1

1. The problem involves factoring a quadratic polynomial.
2. The quadratic polynomial is $Q(x) = 6x^2 + 7x + 1$ .
3. The polynomial is partially factored as $(6x + 1)(\square)$ .
4. We need to find the missing factor.

STEP 2

1. Identify the structure of the quadratic polynomial.
2. Use the given partial factorization to find the missing factor.
3. Verify the complete factorization.

STEP 3

Identify the structure of the quadratic polynomial $Q(x) = 6x^2 + 7x + 1$ .
The polynomial is a quadratic in the form $ax^2 + bx + c$ , where $a = 6$ , $b = 7$ , and $c = 1$ .

STEP 4

Use the given partial factorization $(6x + 1)(\square)$ to find the missing factor.
To find the missing factor, we can set up the equation:
$(6x + 1)(\square) = 6x^2 + 7x + 1$
Let the missing factor be $(x + b)$ . Then:
$(6x + 1)(x + b) = 6x^2 + 7x + 1$
Expand the left side:
$6x^2 + 6bx + x + b = 6x^2 + 7x + 1$

STEP 5

Match coefficients to find $b$ .
From the expanded form:
1. The coefficient of $x$ is $6b + 1 = 7$ .
2. The constant term is $b = 1$ .
Solve for $b$ :
$6b + 1 = 7$ $6b = 6$ $b = 1$
Thus, the missing factor is $(x + 1)$ .

STEP 6

Verify the complete factorization.
Substitute $b = 1$ back into the factorization:
$(6x + 1)(x + 1) = 6x^2 + 6x + x + 1 = 6x^2 + 7x + 1$
The factorization is verified as correct.
The complete factorization of $Q(x) = 6x^2 + 7x + 1$ is $(6x + 1)(x + 1)$ .

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QuestionStep 1 need to divide. However, we note that the denominator can be factored as follows. $\begin{aligned} Q(x) & =6 x^{2}+7 x+1 \\ & =(6 x+1)(\square) \end{aligned}$ Submit Skip (you cannot come back)

Studdy Solution

STEP 1

1. The problem involves factoring a quadratic polynomial.
2. The quadratic polynomial is $Q(x) = 6x^2 + 7x + 1$ .
3. The polynomial is partially factored as $(6x + 1)(\square)$ .
4. We need to find the missing factor.

STEP 2

1. Identify the structure of the quadratic polynomial.
2. Use the given partial factorization to find the missing factor.
3. Verify the complete factorization.

STEP 3

Identify the structure of the quadratic polynomial $Q(x) = 6x^2 + 7x + 1$ .
The polynomial is a quadratic in the form $ax^2 + bx + c$ , where $a = 6$ , $b = 7$ , and $c = 1$ .

STEP 4

STEP 5

Match coefficients to find $b$ .
From the expanded form:
1. The coefficient of $x$ is $6b + 1 = 7$ .
2. The constant term is $b = 1$ .
Solve for $b$ :
$6b + 1 = 7$ $6b = 6$ $b = 1$
Thus, the missing factor is $(x + 1)$ .

STEP 6

Verify the complete factorization.
Substitute $b = 1$ back into the factorization:
$(6x + 1)(x + 1) = 6x^2 + 6x + x + 1 = 6x^2 + 7x + 1$
The factorization is verified as correct.
The complete factorization of $Q(x) = 6x^2 + 7x + 1$ is $(6x + 1)(x + 1)$ .

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QuestionStep 1 need to divide. However, we note that the denominator can be factored as follows. Q(x)=6x2+7x+1=(6x+1)(□)\begin{aligned} Q(x) & =6 x^{2}+7 x+1 \\ & =(6 x+1)(\square) \end{aligned}Q(x)​=6x2+7x+1=(6x+1)(□)​ Submit Skip (you cannot come back)

Studdy Solution

STEP 1

1. The problem involves factoring a quadratic polynomial.2. The quadratic polynomial is Q(x)=6x2+7x+1 Q(x) = 6x^2 + 7x + 1 Q(x)=6x2+7x+1.3. The polynomial is partially factored as (6x+1)(□) (6x + 1)(\square) (6x+1)(□).4. We need to find the missing factor.

STEP 2

1. Identify the structure of the quadratic polynomial.2. Use the given partial factorization to find the missing factor.3. Verify the complete factorization.

STEP 3

Identify the structure of the quadratic polynomial Q(x)=6x2+7x+1 Q(x) = 6x^2 + 7x + 1 Q(x)=6x2+7x+1.The polynomial is a quadratic in the form ax2+bx+c ax^2 + bx + c ax2+bx+c, where a=6 a = 6 a=6, b=7 b = 7 b=7, and c=1 c = 1 c=1.

STEP 4

STEP 5

Match coefficients to find b b b.From the expanded form:1. The coefficient of x x x is 6b+1=7 6b + 1 = 7 6b+1=7.2. The constant term is b=1 b = 1 b=1.Solve for b b b:6b+1=7 6b + 1 = 7 6b+1=7 6b=6 6b = 6 6b=6 b=1 b = 1 b=1Thus, the missing factor is (x+1) (x + 1) (x+1).

STEP 6

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QuestionStep 1 need to divide. However, we note that the denominator can be factored as follows. $\begin{aligned} Q(x) & =6 x^{2}+7 x+1 \\ & =(6 x+1)(\square) \end{aligned}$ Submit Skip (you cannot come back)

1. The problem involves factoring a quadratic polynomial.
2. The quadratic polynomial is $Q(x) = 6x^2 + 7x + 1$ .
3. The polynomial is partially factored as $(6x + 1)(\square)$ .
4. We need to find the missing factor.

1. Identify the structure of the quadratic polynomial.
2. Use the given partial factorization to find the missing factor.
3. Verify the complete factorization.

Identify the structure of the quadratic polynomial $Q(x) = 6x^2 + 7x + 1$ .
The polynomial is a quadratic in the form $ax^2 + bx + c$ , where $a = 6$ , $b = 7$ , and $c = 1$ .

Match coefficients to find $b$ .
From the expanded form:
1. The coefficient of $x$ is $6b + 1 = 7$ .
2. The constant term is $b = 1$ .
Solve for $b$ :
$6b + 1 = 7$ $6b = 6$ $b = 1$
Thus, the missing factor is $(x + 1)$ .