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Answer :
To solve the polynomial expression [tex]\(20x^3 + 15x^5 + 35x^4 - 5x\)[/tex], we want to factor it completely. Follow these steps:
1. Identify the Greatest Common Factor (GCF):
- Look at all the terms: [tex]\(20x^3\)[/tex], [tex]\(15x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(-5x\)[/tex].
- The coefficients are 20, 15, 35, and -5. The GCF of these numbers is 5.
- The variable part is [tex]\(x\)[/tex], and the smallest exponent is 1. So, the GCF for the variables is [tex]\(x\)[/tex].
- Therefore, the overall GCF of the entire expression is [tex]\(5x\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF [tex]\(5x\)[/tex]:
- [tex]\( \frac{20x^3}{5x} = 4x^2 \)[/tex]
- [tex]\( \frac{15x^5}{5x} = 3x^4 \)[/tex]
- [tex]\( \frac{35x^4}{5x} = 7x^3 \)[/tex]
- [tex]\( \frac{-5x}{5x} = -1 \)[/tex]
- So, the expression becomes:
[tex]\[
5x(3x^4 + 7x^3 + 4x^2 - 1)
\][/tex]
3. Verify if further factoring is possible:
- Attempt to factor the polynomial [tex]\(3x^4 + 7x^3 + 4x^2 - 1\)[/tex] further, if possible. However, this polynomial doesn't factor neatly with rational coefficients, so we will keep it as is.
Therefore, the factored form of the polynomial [tex]\(20x^3 + 15x^5 + 35x^4 - 5x\)[/tex] is:
[tex]\[
5x(3x^4 + 7x^3 + 4x^2 - 1)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
- Look at all the terms: [tex]\(20x^3\)[/tex], [tex]\(15x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(-5x\)[/tex].
- The coefficients are 20, 15, 35, and -5. The GCF of these numbers is 5.
- The variable part is [tex]\(x\)[/tex], and the smallest exponent is 1. So, the GCF for the variables is [tex]\(x\)[/tex].
- Therefore, the overall GCF of the entire expression is [tex]\(5x\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF [tex]\(5x\)[/tex]:
- [tex]\( \frac{20x^3}{5x} = 4x^2 \)[/tex]
- [tex]\( \frac{15x^5}{5x} = 3x^4 \)[/tex]
- [tex]\( \frac{35x^4}{5x} = 7x^3 \)[/tex]
- [tex]\( \frac{-5x}{5x} = -1 \)[/tex]
- So, the expression becomes:
[tex]\[
5x(3x^4 + 7x^3 + 4x^2 - 1)
\][/tex]
3. Verify if further factoring is possible:
- Attempt to factor the polynomial [tex]\(3x^4 + 7x^3 + 4x^2 - 1\)[/tex] further, if possible. However, this polynomial doesn't factor neatly with rational coefficients, so we will keep it as is.
Therefore, the factored form of the polynomial [tex]\(20x^3 + 15x^5 + 35x^4 - 5x\)[/tex] is:
[tex]\[
5x(3x^4 + 7x^3 + 4x^2 - 1)
\][/tex]
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