Problem 54 Factor each trinomial completely... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 6: Problem 54

Factor each trinomial completely. $$ 4 x^{4}+2 x^{3}+x^{2} $$

Short Answer

Expert verified

The expression factors to $x^2 (4x^2 + 2x + 1)$.

Step by step solution

- Identify a common factor

First, observe that each term in the trinomial shares a common factor. In this case, notice that the variable and its powers can be factored out. The common factor among the terms is $x^2$.

- Factor out the common term

Factor $x^2$ out of each term in the trinomial: \[4 x^4 + 2 x^3 + x^2 = x^2 (4 x^2 + 2 x + 1)\]. Now we have the expression $x^2 (4x^2 + 2x + 1)$.

- Check for further factorization

Examine the quadratic expression $4x^2 + 2x + 1$ within the parentheses for further factorization. However, this quadratic expression cannot be factored into simpler binomials using integer coefficients.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Trinomials

Factoring trinomials involves breaking down a polynomial with three terms into simpler components that, when multiplied together, give the original trinomial. This process simplifies complex expressions, making them easier to manipulate and solve. To factor a trinomial, one looks for patterns or common factors that can be factored out.
For example, in the trinomial $4 x^{4}+2x^{3}+x^{2}$, we initially look for a common factor present in each term.

Common Factor

A common factor is a term that divides all the terms in a polynomial. To identify a common factor, you should look for the greatest common divisor (GCD) that each term shares.
In our example, $4 x^{4}+2x^{3}+x^{2}$, each term includes $x^{2}$ as a factor: $4 x^{4}, 2 x^{3}, \text{and } x^{2}$. By factoring out $x^{2}$, we simplify the polynomial, reducing it to $x^{2} (4 x^{2} + 2 x + 1)$. This makes further operations on the polynomial more straightforward.

Quadratic Expression

A quadratic expression is a polynomial where the highest exponent of the variable is 2. These can typically be written in the form ax^2 + bx + c. Some quadratics can be factored further, but only if they have particular forms or factorable products of their coefficients.
In the given factorized form $4 x^{2} + 2 x + 1$, each component does not fit simpler binomial forms, meaning no integer solutions are available for further factorization. Therefore, the reduced form $x^{2} (4 x^{2} + 2 x + 1)$ is its simplest factored state.
Understanding how to manage and factor quadratics is fundamental in algebra, paving the way for solving more complex equations and functions.

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91影视

Factor each trinomial completely. $$ 4 x^{4}+2 x^{3}+x^{2} $$

Short Answer

Step by step solution

- Identify a common factor

- Factor out the common term

- Check for further factorization

Key Concepts

Factoring Trinomials

Common Factor

Quadratic Expression

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