Math / Algebra

QuestionFactor completely. If a polynomial is prime, state this. $20 x^{3}-5 x$
Select the correct choice below and, if necessary, fill in the answer box within your choice. A. $20 x^{3}-5 x=$ $\square$ B. The polynomial is prime.

Studdy Solution

STEP 1

1. The polynomial given is $20x^3 - 5x$ .
2. We need to factor the polynomial completely.
3. A polynomial is considered prime if it cannot be factored further over the integers.

STEP 2

1. Identify and factor out the greatest common factor (GCF) from the polynomial.
2. Check if the resulting expression can be factored further.

STEP 3

Identify the greatest common factor (GCF) of the terms in the polynomial $20x^3 - 5x$ .
- The GCF of the coefficients 20 and 5 is 5. - The GCF of the variables $x^3$ and $x$ is $x$ .
Therefore, the GCF of the polynomial is $5x$ .

STEP 4

Factor out the GCF $5x$ from the polynomial:
$20x^3 - 5x = 5x(4x^2 - 1)$

STEP 5

Check if the expression inside the parentheses $4x^2 - 1$ can be factored further.
- Recognize $4x^2 - 1$ as a difference of squares:
$4x^2 - 1 = (2x)^2 - 1^2$
- Apply the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$ .
$(2x)^2 - 1^2 = (2x - 1)(2x + 1)$

STEP 6

Substitute the factored form of $4x^2 - 1$ back into the expression:
$20x^3 - 5x = 5x(2x - 1)(2x + 1)$
This is the completely factored form of the polynomial.
The correct choice is:
A. $20x^3 - 5x = 5x(2x - 1)(2x + 1)$

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QuestionFactor completely. If a polynomial is prime, state this. $20 x^{3}-5 x$
Select the correct choice below and, if necessary, fill in the answer box within your choice. A. $20 x^{3}-5 x=$ $\square$ B. The polynomial is prime.

Studdy Solution

STEP 1

1. The polynomial given is $20x^3 - 5x$ .
2. We need to factor the polynomial completely.
3. A polynomial is considered prime if it cannot be factored further over the integers.

STEP 2

1. Identify and factor out the greatest common factor (GCF) from the polynomial.
2. Check if the resulting expression can be factored further.

STEP 3

Identify the greatest common factor (GCF) of the terms in the polynomial $20x^3 - 5x$ .
- The GCF of the coefficients 20 and 5 is 5. - The GCF of the variables $x^3$ and $x$ is $x$ .
Therefore, the GCF of the polynomial is $5x$ .

STEP 4

Factor out the GCF $5x$ from the polynomial:
$20x^3 - 5x = 5x(4x^2 - 1)$

STEP 5

Check if the expression inside the parentheses $4x^2 - 1$ can be factored further.
- Recognize $4x^2 - 1$ as a difference of squares:
$4x^2 - 1 = (2x)^2 - 1^2$
- Apply the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$ .
$(2x)^2 - 1^2 = (2x - 1)(2x + 1)$

STEP 6

Substitute the factored form of $4x^2 - 1$ back into the expression:
$20x^3 - 5x = 5x(2x - 1)(2x + 1)$
This is the completely factored form of the polynomial.
The correct choice is:
A. $20x^3 - 5x = 5x(2x - 1)(2x + 1)$

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QuestionFactor completely. If a polynomial is prime, state this. 20x3−5x20 x^{3}-5 x20x3−5xSelect the correct choice below and, if necessary, fill in the answer box within your choice. A. 20x3−5x=20 x^{3}-5 x=20x3−5x= □\square□ B. The polynomial is prime.

Studdy Solution

STEP 1

1. The polynomial given is 20x3−5x20x^3 - 5x20x3−5x.2. We need to factor the polynomial completely.3. A polynomial is considered prime if it cannot be factored further over the integers.

STEP 2

1. Identify and factor out the greatest common factor (GCF) from the polynomial.2. Check if the resulting expression can be factored further.

STEP 3

Identify the greatest common factor (GCF) of the terms in the polynomial 20x3−5x20x^3 - 5x20x3−5x.- The GCF of the coefficients 20 and 5 is 5. - The GCF of the variables x3x^3x3 and xxx is xxx.Therefore, the GCF of the polynomial is 5x5x5x.

STEP 4

Factor out the GCF 5x5x5x from the polynomial:20x3−5x=5x(4x2−1) 20x^3 - 5x = 5x(4x^2 - 1) 20x3−5x=5x(4x2−1)

STEP 5

STEP 6

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QuestionFactor completely. If a polynomial is prime, state this. $20 x^{3}-5 x$
Select the correct choice below and, if necessary, fill in the answer box within your choice. A. $20 x^{3}-5 x=$ $\square$ B. The polynomial is prime.

1. The polynomial given is $20x^3 - 5x$ .
2. We need to factor the polynomial completely.
3. A polynomial is considered prime if it cannot be factored further over the integers.

1. Identify and factor out the greatest common factor (GCF) from the polynomial.
2. Check if the resulting expression can be factored further.

Identify the greatest common factor (GCF) of the terms in the polynomial $20x^3 - 5x$ .
- The GCF of the coefficients 20 and 5 is 5. - The GCF of the variables $x^3$ and $x$ is $x$ .
Therefore, the GCF of the polynomial is $5x$ .

Factor out the GCF $5x$ from the polynomial:
$20x^3 - 5x = 5x(4x^2 - 1)$