Answer :
- Factor out the greatest common factor: $7x(4x^3 + 16x^2 - 3x - 12)$.
- Factor by grouping: $7x(x+4)(4x^2-3)$.
- The factored form of the polynomial is $\boxed{7x(x + 4)(4x^2 - 3)}$.
### Explanation
1. Problem Analysis
We are given the polynomial $28 x^4+112 x^3-21 x^2-84 x$ and we want to factor it completely.
2. Factoring out the GCF
First, we can factor out the greatest common factor (GCF) from all the terms. The GCF of the coefficients is 7, and each term has at least one factor of $x$. So, we can factor out $7x$ from the polynomial:
$$28 x^4+112 x^3-21 x^2-84 x = 7x(4x^3 + 16x^2 - 3x - 12)$$
3. Factoring by Grouping
Now, we need to factor the cubic polynomial $4x^3 + 16x^2 - 3x - 12$. We can try factoring by grouping. Group the first two terms and the last two terms:
$$(4x^3 + 16x^2) + (-3x - 12)$$
Factor out the GCF from each group:
$$4x^2(x + 4) - 3(x + 4)$$
Now, we can factor out the common binomial factor $(x + 4)$:
$$(4x^2 - 3)(x + 4)$$
4. Complete Factorization
So, the factored form of the cubic polynomial is $(4x^2 - 3)(x + 4)$. Therefore, the complete factorization of the given polynomial is:
$$7x(4x^2 - 3)(x + 4)$$
We can further factor $4x^2 - 3$ as a difference of squares. Notice that $4x^2 - 3 = (2x)^2 - (\sqrt{3})^2$. Thus,
$$4x^2 - 3 = (2x - \sqrt{3})(2x + \sqrt{3})$$
So the complete factorization is:
$$7x(x+4)(2x - \sqrt{3})(2x + \sqrt{3})$$
5. Final Answer
The factored form of the polynomial $28 x^4+112 x^3-21 x^2-84 x$ is $7x(x + 4)(4x^2 - 3)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and engineering. For example, in physics, you might need to find the roots of a polynomial to determine the equilibrium points of a system. In engineering, factoring polynomials can help simplify complex equations when designing structures or circuits. Factoring also helps in simplifying expressions and solving equations, making it a crucial tool in various fields.
- Factor by grouping: $7x(x+4)(4x^2-3)$.
- The factored form of the polynomial is $\boxed{7x(x + 4)(4x^2 - 3)}$.
### Explanation
1. Problem Analysis
We are given the polynomial $28 x^4+112 x^3-21 x^2-84 x$ and we want to factor it completely.
2. Factoring out the GCF
First, we can factor out the greatest common factor (GCF) from all the terms. The GCF of the coefficients is 7, and each term has at least one factor of $x$. So, we can factor out $7x$ from the polynomial:
$$28 x^4+112 x^3-21 x^2-84 x = 7x(4x^3 + 16x^2 - 3x - 12)$$
3. Factoring by Grouping
Now, we need to factor the cubic polynomial $4x^3 + 16x^2 - 3x - 12$. We can try factoring by grouping. Group the first two terms and the last two terms:
$$(4x^3 + 16x^2) + (-3x - 12)$$
Factor out the GCF from each group:
$$4x^2(x + 4) - 3(x + 4)$$
Now, we can factor out the common binomial factor $(x + 4)$:
$$(4x^2 - 3)(x + 4)$$
4. Complete Factorization
So, the factored form of the cubic polynomial is $(4x^2 - 3)(x + 4)$. Therefore, the complete factorization of the given polynomial is:
$$7x(4x^2 - 3)(x + 4)$$
We can further factor $4x^2 - 3$ as a difference of squares. Notice that $4x^2 - 3 = (2x)^2 - (\sqrt{3})^2$. Thus,
$$4x^2 - 3 = (2x - \sqrt{3})(2x + \sqrt{3})$$
So the complete factorization is:
$$7x(x+4)(2x - \sqrt{3})(2x + \sqrt{3})$$
5. Final Answer
The factored form of the polynomial $28 x^4+112 x^3-21 x^2-84 x$ is $7x(x + 4)(4x^2 - 3)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and engineering. For example, in physics, you might need to find the roots of a polynomial to determine the equilibrium points of a system. In engineering, factoring polynomials can help simplify complex equations when designing structures or circuits. Factoring also helps in simplifying expressions and solving equations, making it a crucial tool in various fields.
