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Chapter 5: Problem 62

Factor completely. \(x^{3}-7 x^{2}-4 x+28\)

Short Answer

Expert verified
The factored form is (x - 7)(x - 2)(x + 2).

Step by step solution

01

- Group terms

Group the polynomial in pairs to facilitate factoring by grouping. Rewrite the expression:o x^3 - 7x^2 - 4x + 28 = (x^3 - 7x^2) + (-4x + 28).
02

- Factor out the greatest common factor (GCF) from each group

Look for the greatest common factor (GCF) in each group. In the first group, the GCF is x^2.In the second group, the GCF is -4. Factoring out these GCFs from each pair, we get:x^2(x - 7) - 4(x - 7).
03

- Factor out the common binomial factor

Notice that (x - 7) is a common factor in both terms. Factor out the (x - 7) from the expression:(x - 7)(x^2 - 4).
04

- Factor the remaining quadratic expression

Recognize that the remaining quadratic expression, x^2 - 4 is a difference of squares. Recall that a^2 - b^2 = (a - b)(a + b). Apply this formula to factor x^2 - 4:(x - 2)(x + 2).
05

- Combine all factors

Now, combine all the factors obtained in the previous steps to get the complete factored form of the given polynomial:(x - 7)(x - 2)(x + 2).

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Key Concepts

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

Factoring by Grouping
Factoring by grouping is a technique used to factor polynomials that have four terms. This method works by grouping terms with common factors.
For example, consider the polynomial: ewline x^3 - 7x^2 - 4x + 28 ewline Group the terms in pairs: (x^3 - 7x^2) and (-4x + 28).
Once grouped, we can easily factor each group separately by identifying common factors.
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest factor shared by a set of numbers. In factoring, finding the GCF simplifies an expression.
For instance, in our example: (x^3 - 7x^2) and (-4x + 28), the GCF of the first group is x^2 and the GCF of the second group is -4.
This allows us to rewrite the polynomial as: ewline x^2(x - 7) - 4(x - 7).
Once the GCFs are factored out, we can further simplify the expression.
Difference of Squares
A difference of squares is a specific polynomial form written as: a^2 - b^2.
This can be factored using the formula: ewline (a - b)(a + b).
In our polynomial example, ewline x^2 - 4 is a difference of squares, where a = x and b = 2.
Applying the formula, we get: (x - 2)(x + 2).
Factored Form
The factored form of a polynomial is an expression that represents the polynomial as a product of its factors.
Factoring helps in simplifying polynomials and solving polynomial equations.
After factoring our polynomial completely, it鈥檚 written as: ewline (x - 7)(x - 2)(x + 2).
This is the factored form of the polynomial x^3 - 7x^2 - 4x + 28.
Quadratic Expressions
Quadratic expressions are polynomials of degree 2 and have the form: ewline ax^2 + bx + c.
When factoring quadratic expressions, look for patterns such as: ewline - Perfect square trinomials - Differences of squares - Simple quadratics ewline In our example, after factoring by grouping, we encountered x^2 - 4, which is a quadratic expression that we identified as a difference of squares.

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