Problem 73 Factor completely. \(x^{3}+5 x... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 73

Factor completely. $x^{3}+5 x^{2}-4 x-20$

Short Answer

Expert verified

(x + 5)(x + 2)(x - 2)

Step by step solution

Group Terms

Group the polynomial into pairs to facilitate factoring by grouping. Split the terms as follows: o(x^3 + 5x^2) - (4x + 20).

Factor Out the GCD from Each Pair

Find the greatest common divisor (GCD) of each pair and factor them out:o x^2(x + 5) - 4(x + 5).

Factor Using the Common Binomial

Notice that o(x + 5) is a common factor. Factor it out:o(x + 5)(x^2 - 4).

Factor the Difference of Squares

Factor the remaining expression, which is a difference of squares:o x^2 - 4 = (x + 2)(x - 2). Thus, the factorization becomes o(x + 5)(x + 2)(x - 2).

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Key Concepts

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

Grouping Terms

Grouping terms is a method used to simplify a polynomial by creating smaller groups that can be factored separately.
This is especially useful in higher-degree polynomials where direct factoring isn't obvious.
In our exercise, we started with the polynomial $x^3 + 5x^2 - 4x - 20$.
By grouping, we made it more manageable:

First group: $x^3 + 5x^2$
Second group: $-4x - 20$

The primary goal of grouping is to prepare each part in such a way that further factors can be found more easily.
This step sets the stage for discovering common factors in the following steps.

Greatest Common Divisor (GCD)

The greatest common divisor is the largest number or polynomial that can evenly divide each term in a set of terms.
In simpler terms, it鈥檚 the highest factor shared among the terms.

In our grouped polynomial, the first group $x^3 + 5x^2$ and the second group $-4x - 20$ both have common factors:

GCD of $x^3$ and $5x^2$ is $x^2$.
GCD of $-4x$ and $-20$ is $ -4$.

By factoring these out, we get:
$x^2(x + 5) - 4(x + 5)$.
This reveals a common binomial factor, needed for further simplification.

Difference of Squares

The difference of squares is a specific type of polynomial: $a^2 - b^2$.
This can always be factored into $(a + b)(a - b)$.

For our exercise, we identified $x^2 - 4$:

Recognize $4$ as $2^2$, making the polynomial fit the $a^2 - b^2$ form.
Factor it as $(x + 2)(x - 2)$.

.
Combining this with our previously factored expression, we get
$(x + 5)(x + 2)(x - 2)$.

Using the difference of squares simplifies complex polynomials into easy-to-manage binomials.

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91影视

Factor completely. \(x^{3}+5 x^{2}-4 x-20\)

Short Answer

Step by step solution

Group Terms

Factor Out the GCD from Each Pair

Factor Using the Common Binomial

Factor the Difference of Squares

Key Concepts

Grouping Terms

Greatest Common Divisor (GCD)

Difference of Squares

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