How do you factor the polynomial x^4 + 7x^3 + 12x^2 and find the GCF?

Understand the Problem

The question is discussing various methods for factoring polynomials and finding the greatest common factor (GCF). It presents a specific polynomial and breaks down the steps for factoring it, showing the process of identification and simplification.

Answer

The factored form of the polynomial is $x^2 (x + 3)(x + 4)$.

Steps to Solve

Identify the polynomial and its components

The given polynomial is $x^4 + 7x^3 + 12x^2$. We will factor out the greatest common factor (GCF) first.
Find the GCF

The GCF of the terms $x^4$, $7x^3$, and $12x^2$ is $x^2$, as it is the highest power of $x$ present in all terms.
Factor out the GCF

Factor out $x^2$ from the polynomial:

$$ x^4 + 7x^3 + 12x^2 = x^2(x^2 + 7x + 12) $$
Factor the quadratic expression

Now we will factor the quadratic $x^2 + 7x + 12$. We look for two numbers that multiply to $12$ (the constant term) and add to $7$ (the coefficient of $x$). The numbers $3$ and $4$ work.
Write the factored form of the quadratic

Thus, we can write:

$$ x^2 + 7x + 12 = (x + 3)(x + 4) $$
Combine all factors

Now combine all the factors we have:

$$ x^4 + 7x^3 + 12x^2 = x^2(x + 3)(x + 4) $$
Identify if factors are prime

The factors $x^2$, $(x + 3)$, and $(x + 4)$ are considered prime (irreducible) over the real numbers.

The factored form of the polynomial $x^4 + 7x^3 + 12x^2$ is

$$ x^2 (x + 3)(x + 4) $$

More Information

Factoring polynomials is an essential skill in algebra, used to simplify expressions and solve equations. The GCF approach helps break down complex polynomials into simpler components, making them easier to work with.

Tips

Forgetting to factor out the GCF first can lead to complicated expressions.
Misidentifying the pairs of numbers that multiply to the constant term and add to the middle coefficient.

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