Problem 51 Factor completely each of the po... [FREE SOLUTION]

Chapter 3: Problem 51

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{4}-9 x^{2}+8$$

Short Answer

Expert verified

The polynomial can be factored as $(x^2 - 8)(x - 1)(x + 1)$.

Step by step solution

Recognize the Type of Polynomial

The given polynomial is a quadratic polynomial in terms of $x^2$. It can be written as $(x^2)^2 - 9(x^2) + 8$, which shows it's quadratic in form $aX^2 + bX + c$ where $X = x^2$.

Identify a Method for Factoring

We can use the factoring method for quadratic equations. We aim to write the polynomial in the form $ (x^2 + m)(x^2 + n) $ to match $x^4 - 9x^2 + 8$.

Use the AC Method for Quadratic Factoring

Here, $a = 1$, $b = -9$, and $c = 8$. Use the AC method: multiply $a$ and $c$ to get 8. We need two numbers that multiply to 8 and add to $-9$. These numbers are $-1$ and $-8$.

Factor the Quadratic

Rewrite the middle term $-9x^2$ as $-x^2 - 8x^2$. The polynomial now looks like: $x^4 - 1x^2 - 8x^2 + 8$. Factor by grouping: $(x^4 - x^2) - (8x^2 - 8)$. Factor each group: $x^2(x^2 - 1) - 8(x^2 - 1)$. This gives us $(x^2 - 8)(x^2 - 1)$.

Check for Further Factorability

Factor $x^2 - 1$ as it is a difference of squares: $(x - 1)(x + 1)$. So the full factorization is $(x^2 - 8)(x - 1)(x + 1)$. $x^2 - 8$ isn't further factorable over the integers.

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Key Concepts

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

quadratic polynomial

A quadratic polynomial is an expression that involves a variable raised to the second power. Typically, its standard form is represented as $ ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants. In our case, the polynomial given can be identified in a slightly different form, as $ (x^2)^2 - 9(x^2) + 8 $, which is quadratic in terms of $ x^2 $. This conversion is key since it recognizes the polynomial as a quadratic function by temporarily substituting $ X = x^2 $, thereby simplifying it to the form $ aX^2 + bX + c $. Recognizing this form makes it easier to apply standard factoring techniques associated with quadratic equations.

factoring methods

Factoring is a process of breaking down a complex expression into a product of simpler expressions. There are various factoring methods used to simplify polynomials, such as:

Factoring by grouping
Using the difference of squares identity
Leveling through trinomial factorization like $ aX^2 + bX + c $

Each of these methods is useful for specific types of polynomials. For a quadratic polynomial, we often seek two numbers that multiply to the product of $ a $ and $ c $ and add up to $ b $. Recognizing the right method to apply is crucial in polynomial factoring, as it simplifies complex polynomials, making them more manageable to work with and understand.

difference of squares

The difference of squares is a specific form of factoring used when a polynomial expresses a subtraction between two squared terms. It is built upon the identity $ a^2 - b^2 = (a - b)(a + b) $. This nifty tool allows us to factor expressions like $ x^2 - 1 $ as $(x - 1)(x + 1)$. In the step-by-step solution, when we reached $ x^2 - 1 $, we utilized this identity to break it down into two linear factors. Recognizing each segment of a polynomial that fits this pattern helps in successive simplification, breaking down compound expressions into multiple, straightforward components.

AC method

The AC method, known as the "product-sum" method, is particularly handy for factoring quadratic expressions. It involves a few distinct steps:

First, identify $ a $, $ b $, and $ c $ in the quadratic polynomial $ ax^2 + bx + c $.
Calculate $ a \times c $ (this product is key for the method), and find two numbers that multiply to $ ac $ and sum to $ b $.
Split the middle term using these two numbers, transforming it into two separate terms.
Following that, factor by grouping.

For example, in our problem, the polynomial could be transformed by rewriting $-9x^2$ as $-x^2 - 8x^2$, using the numbers $-1$ and $-8$. It's a strategic method that decomposes the problem into simpler factoring steps, ultimately leading to the correct factorization.

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91影视

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{4}-9 x^{2}+8$$

Short Answer

Step by step solution

Recognize the Type of Polynomial

Identify a Method for Factoring

Use the AC Method for Quadratic Factoring

Factor the Quadratic

Check for Further Factorability

Key Concepts

quadratic polynomial

factoring methods

difference of squares

AC method

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