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Answer :
To factor the greatest common factor (GCF) from the polynomial [tex]\(14x^5 - 35x^4 - 28x^3\)[/tex], follow these steps:
1. Identify the GCF of the coefficients: Look at the numerical coefficients of the polynomial's terms: 14, 35, and 28. The greatest common factor of these numbers is 7.
2. Identify the GCF of the variable terms: All terms in the polynomial include [tex]\(x\)[/tex]. The smallest power of [tex]\(x\)[/tex] among the terms is [tex]\(x^3\)[/tex]. This means the GCF for the variable part is [tex]\(x^3\)[/tex].
3. Combine the GCFs: The overall greatest common factor for the entire polynomial is [tex]\(7x^3\)[/tex].
4. Factor the polynomial: Divide each term in the polynomial by the GCF and write down the result:
- Divide [tex]\(14x^5\)[/tex] by [tex]\(7x^3\)[/tex] to get [tex]\(2x^2\)[/tex].
- Divide [tex]\(-35x^4\)[/tex] by [tex]\(7x^3\)[/tex] to get [tex]\(-5x\)[/tex].
- Divide [tex]\(-28x^3\)[/tex] by [tex]\(7x^3\)[/tex] to get [tex]\(-4\)[/tex].
5. Write the factored form: Combine the GCF with the result from step 4 inside parentheses to express the factored polynomial. The factored form is:
[tex]\[
7x^3(2x^2 - 5x - 4)
\][/tex]
That's how you factor out the greatest common factor from the polynomial [tex]\(14x^5 - 35x^4 - 28x^3\)[/tex].
1. Identify the GCF of the coefficients: Look at the numerical coefficients of the polynomial's terms: 14, 35, and 28. The greatest common factor of these numbers is 7.
2. Identify the GCF of the variable terms: All terms in the polynomial include [tex]\(x\)[/tex]. The smallest power of [tex]\(x\)[/tex] among the terms is [tex]\(x^3\)[/tex]. This means the GCF for the variable part is [tex]\(x^3\)[/tex].
3. Combine the GCFs: The overall greatest common factor for the entire polynomial is [tex]\(7x^3\)[/tex].
4. Factor the polynomial: Divide each term in the polynomial by the GCF and write down the result:
- Divide [tex]\(14x^5\)[/tex] by [tex]\(7x^3\)[/tex] to get [tex]\(2x^2\)[/tex].
- Divide [tex]\(-35x^4\)[/tex] by [tex]\(7x^3\)[/tex] to get [tex]\(-5x\)[/tex].
- Divide [tex]\(-28x^3\)[/tex] by [tex]\(7x^3\)[/tex] to get [tex]\(-4\)[/tex].
5. Write the factored form: Combine the GCF with the result from step 4 inside parentheses to express the factored polynomial. The factored form is:
[tex]\[
7x^3(2x^2 - 5x - 4)
\][/tex]
That's how you factor out the greatest common factor from the polynomial [tex]\(14x^5 - 35x^4 - 28x^3\)[/tex].
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