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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 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

9 answers

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:

\(x - 2\).

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

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:

\(4x\).

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

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.

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

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.

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 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 \(2x\):

\[ 2x(x^4 - 2x^2 + 1) = 0 \]

  1. Setting the factored equation equal to zero:

\[ 2x(x^4 - 2x^2 + 1) = 0 \]

This gives us the first solution directly:

\[ 2x = 0 \Rightarrow x = 0 \]

  1. 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:

\[ x = 1 \quad \text{or} \quad x = -1 \]

  1. 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:

\(x = -1, x = 0, x = 1\).