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Renee is asked to find a factor of x3−5x2+6x , given that x is one of the linear factors. Which of the following is another cor...Asked by Nanika Zoldyck
Renee is asked to find a factor of x3−5x2+6x
, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)
Responses
x2+5x+6
x squared plus 5 x plus 6
x+3
x plus 3
x−2
x minus 2
x+2
, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)
Responses
x2+5x+6
x squared plus 5 x plus 6
x+3
x plus 3
x−2
x minus 2
x+2
Answers
Answered by
Nanika Zoldyck
Which of the following is the GCF of the polynomial 4x4+12x3−36x2+24x
?(1 point)
Responses
6x
6 x
4
4
2x
2 x
4x
?(1 point)
Responses
6x
6 x
4
4
2x
2 x
4x
Answered by
Nanika Zoldyck
Which of the following is a complete set of solutions to 3x3+9x2−54x=0
?(1 point)
Responses
0, −3
, 6
0, negative 3 , 6
no solutions
no solutions
0
0
0, 3, −6
?(1 point)
Responses
0, −3
, 6
0, negative 3 , 6
no solutions
no solutions
0
0
0, 3, −6
Answered by
Nanika Zoldyck
Which pattern can be used to factor the high-degree polynomial 16x8−49x2
?(1 point)
Responses
Neither the difference of squares nor the perfect square trinomial pattern can be used to factor 16x8−49x2
.
Neither the difference of squares nor the perfect square trinomial pattern can be used to factor 16 x superscript 8 baseline minus 49 x squared .
(a+b)2=a2+2ab+b2
left parenthesis a plus b right parenthesis squared equals a squared plus 2 a b plus b squared
(a−b)(a+b)=a2−b2
left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis equals a squared minus b squared
(a−b)2=a2−2ab+b2
?(1 point)
Responses
Neither the difference of squares nor the perfect square trinomial pattern can be used to factor 16x8−49x2
.
Neither the difference of squares nor the perfect square trinomial pattern can be used to factor 16 x superscript 8 baseline minus 49 x squared .
(a+b)2=a2+2ab+b2
left parenthesis a plus b right parenthesis squared equals a squared plus 2 a b plus b squared
(a−b)(a+b)=a2−b2
left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis equals a squared minus b squared
(a−b)2=a2−2ab+b2
Answered by
Nanika Zoldyck
Factor the high-degree polynomial 2x5−4x3+2x
to solve the polynomial equation 2x5−4x3+2x=0
.(1 point)
Responses
x=−1
, x=0
, x=1
x equals negative 1 , x equals 0 , x equals 1
x=2
x equals 2
x=−1
, x=1
x equals negative 1 , x equals 1
x=0
, x=1
to solve the polynomial equation 2x5−4x3+2x=0
.(1 point)
Responses
x=−1
, x=0
, x=1
x equals negative 1 , x equals 0 , x equals 1
x=2
x equals 2
x=−1
, x=1
x equals negative 1 , x equals 1
x=0
, x=1
Answered by
GPT-4o mini
To find the factors of the polynomial \(x^3 - 5x^2 + 6x\), we can start by factoring out the common factor \(x\):
\[
x(x^2 - 5x + 6)
\]
Next, we need to factor the quadratic \(x^2 - 5x + 6\). We can do this by looking for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-2\) and \(-3\) satisfy these conditions:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
Thus, the complete factorization of \(x^3 - 5x^2 + 6x\) is:
\[
x(x - 2)(x - 3)
\]
The linear factors of the polynomial are \(x\), \(x - 2\), and \(x - 3\).
Given the options:
1. \(x^2 + 5x + 6\)
2. \(x + 3\)
3. \(x - 2\)
4. \(x + 2\)
Among these, the correct factor (another factor of the polynomial) is:
**\(x - 2\)**.
\[
x(x^2 - 5x + 6)
\]
Next, we need to factor the quadratic \(x^2 - 5x + 6\). We can do this by looking for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-2\) and \(-3\) satisfy these conditions:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
Thus, the complete factorization of \(x^3 - 5x^2 + 6x\) is:
\[
x(x - 2)(x - 3)
\]
The linear factors of the polynomial are \(x\), \(x - 2\), and \(x - 3\).
Given the options:
1. \(x^2 + 5x + 6\)
2. \(x + 3\)
3. \(x - 2\)
4. \(x + 2\)
Among these, the correct factor (another factor of the polynomial) is:
**\(x - 2\)**.
Answered by
GPT-4o mini
To find the greatest common factor (GCF) of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\), we can start by identifying the GCF of the coefficients and the GCF of the variable factors.
1. **Coefficients:** The coefficients are \(4\), \(12\), \(-36\), and \(24\). The GCF of these numbers is \(4\).
2. **Variable factors:** Each term contains at least one factor of \(x\):
- \(4x^4\) has \(x^4\)
- \(12x^3\) has \(x^3\)
- \(-36x^2\) has \(x^2\)
- \(24x\) has \(x^1\)
The lowest power of \(x\) in these terms is \(x^1\) (or just \(x\)).
Combining both parts, the GCF of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\) is:
\[
4x
\]
Therefore, the correct response is:
**\(4x\)**.
1. **Coefficients:** The coefficients are \(4\), \(12\), \(-36\), and \(24\). The GCF of these numbers is \(4\).
2. **Variable factors:** Each term contains at least one factor of \(x\):
- \(4x^4\) has \(x^4\)
- \(12x^3\) has \(x^3\)
- \(-36x^2\) has \(x^2\)
- \(24x\) has \(x^1\)
The lowest power of \(x\) in these terms is \(x^1\) (or just \(x\)).
Combining both parts, the GCF of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\) is:
\[
4x
\]
Therefore, the correct response is:
**\(4x\)**.
Answered by
GPT-4o mini
To solve the equation \(3x^3 + 9x^2 - 54x = 0\), we can start by factoring out the greatest common factor (GCF) from all the terms.
1. First, factor out \(3x\):
\[
3x(x^2 + 3x - 18) = 0
\]
2. Now, we set each factor to zero:
- The first factor gives \(3x = 0\), which leads to:
\[
x = 0
\]
- Next, we need to solve the quadratic \(x^2 + 3x - 18 = 0\). We can factor this quadratic equation:
\[
(x + 6)(x - 3) = 0
\]
3. Setting each factor to zero gives us the additional solutions:
- \(x + 6 = 0 \Rightarrow x = -6\)
- \(x - 3 = 0 \Rightarrow x = 3\)
4. Therefore, the complete set of solutions to the equation \(3x^3 + 9x^2 - 54x = 0\) is:
\[
x = 0, \, x = -6, \, x = 3
\]
Thus, the correct response is:
**0, 3, −6**.
1. First, factor out \(3x\):
\[
3x(x^2 + 3x - 18) = 0
\]
2. Now, we set each factor to zero:
- The first factor gives \(3x = 0\), which leads to:
\[
x = 0
\]
- Next, we need to solve the quadratic \(x^2 + 3x - 18 = 0\). We can factor this quadratic equation:
\[
(x + 6)(x - 3) = 0
\]
3. Setting each factor to zero gives us the additional solutions:
- \(x + 6 = 0 \Rightarrow x = -6\)
- \(x - 3 = 0 \Rightarrow x = 3\)
4. Therefore, the complete set of solutions to the equation \(3x^3 + 9x^2 - 54x = 0\) is:
\[
x = 0, \, x = -6, \, x = 3
\]
Thus, the correct response is:
**0, 3, −6**.
Answered by
GPT-4o mini
To determine the correct pattern to factor the polynomial \(16x^8 - 49x^2\), we can first notice that this expression is a difference of squares.
The general form for the difference of squares is:
\[
a^2 - b^2 = (a - b)(a + b)
\]
1. **Identifying \(a\) and \(b\)**:
- We can rewrite the terms:
- \(16x^8\) can be expressed as \((4x^4)^2\).
- \(49x^2\) can be expressed as \((7x)^2\).
2. **Applying the difference of squares pattern**:
- Now we can rewrite the expression \(16x^8 - 49x^2\) as:
\[
(4x^4)^2 - (7x)^2
\]
- This fits the difference of squares pattern where \(a = 4x^4\) and \(b = 7x\).
3. **Factoring the polynomial**:
- Using the difference of squares formula:
\[
16x^8 - 49x^2 = (4x^4 - 7x)(4x^4 + 7x)
\]
Therefore, the correct choice is:
**(a−b)(a+b)=a²−b²**.
The general form for the difference of squares is:
\[
a^2 - b^2 = (a - b)(a + b)
\]
1. **Identifying \(a\) and \(b\)**:
- We can rewrite the terms:
- \(16x^8\) can be expressed as \((4x^4)^2\).
- \(49x^2\) can be expressed as \((7x)^2\).
2. **Applying the difference of squares pattern**:
- Now we can rewrite the expression \(16x^8 - 49x^2\) as:
\[
(4x^4)^2 - (7x)^2
\]
- This fits the difference of squares pattern where \(a = 4x^4\) and \(b = 7x\).
3. **Factoring the polynomial**:
- Using the difference of squares formula:
\[
16x^8 - 49x^2 = (4x^4 - 7x)(4x^4 + 7x)
\]
Therefore, the correct choice is:
**(a−b)(a+b)=a²−b²**.
Answered by
GPT-4o mini
To factor the polynomial \(2x^5 - 4x^3 + 2x\), we start by factoring out the greatest common factor (GCF) from all the terms.
1. **Identify the GCF**: The GCF of the coefficients \(2\), \(-4\), and \(2\) is \(2\), and each term contains at least one factor of \(x\). So, we can factor out \(2x\):
\[
2x(x^4 - 2x^2 + 1) = 0
\]
2. **Setting the factored equation equal to zero**:
\[
2x(x^4 - 2x^2 + 1) = 0
\]
This gives us the first solution directly:
\[
2x = 0 \Rightarrow x = 0
\]
3. **Now factor the quartic polynomial \(x^4 - 2x^2 + 1\)**. We can rewrite it as a quadratic in terms of \(y = x^2\):
\[
y^2 - 2y + 1 = 0
\]
This factors neatly as:
\[
(y - 1)^2 = 0
\]
So, substituting back for \(x^2\):
\[
(x^2 - 1)(x^2 - 1) = 0 \Rightarrow (x - 1)(x + 1) = 0
\]
This gives:
\[
x = 1 \quad \text{or} \quad x = -1
\]
4. **Collecting all solutions**:
Thus, the complete set of solutions for the equation \(2x^5 - 4x^3 + 2x = 0\) is:
\[
x = 0, \quad x = 1, \quad x = -1
\]
So the correct response is:
**\(x = -1, x = 0, x = 1\)**.
1. **Identify the GCF**: The GCF of the coefficients \(2\), \(-4\), and \(2\) is \(2\), and each term contains at least one factor of \(x\). So, we can factor out \(2x\):
\[
2x(x^4 - 2x^2 + 1) = 0
\]
2. **Setting the factored equation equal to zero**:
\[
2x(x^4 - 2x^2 + 1) = 0
\]
This gives us the first solution directly:
\[
2x = 0 \Rightarrow x = 0
\]
3. **Now factor the quartic polynomial \(x^4 - 2x^2 + 1\)**. We can rewrite it as a quadratic in terms of \(y = x^2\):
\[
y^2 - 2y + 1 = 0
\]
This factors neatly as:
\[
(y - 1)^2 = 0
\]
So, substituting back for \(x^2\):
\[
(x^2 - 1)(x^2 - 1) = 0 \Rightarrow (x - 1)(x + 1) = 0
\]
This gives:
\[
x = 1 \quad \text{or} \quad x = -1
\]
4. **Collecting all solutions**:
Thus, the complete set of solutions for the equation \(2x^5 - 4x^3 + 2x = 0\) is:
\[
x = 0, \quad x = 1, \quad x = -1
\]
So the correct response is:
**\(x = -1, x = 0, x = 1\)**.