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Chapter 5: Problem 65

The following exercises are of mixed variety. Factor each polynomial. $$ 21 m^{4}-32 m^{2}-5 $$

Short Answer

Expert verified
The factored form of is .

Step by step solution

01

Identify the polynomial structure

Observe that the polynomial is a quadratic in form, represented as: is a quadratic expression where: - - - Identify the polynomial structure and write it as a quadratic expression: Let So the equation becomes: which simplifies to:
02

Apply the quadratic factoring method

Notice that the quadratic expression needs to be factored. For that, we need to find values of and such that: and Let's find such values: By trial and error or the quadratic formula, determine that and this can be verified as: Factor the quadratic: So the expression can be written as:
03

Substitute back the original variable

Replace back with the original variable Therefore, the polynomial can be factored as: Thus, the factorization of the given polynomial is achieved.

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Key Concepts

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

quadratic expression
In mathematics, a quadratic expression is a polynomial of degree 2. It typically takes the form: \[ax^2 + bx + c\] where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

In the given problem, the polynomial \(21 m^{4} - 32 m^{2} - 5\) is structured like a quadratic expression but with different variables. To see this clearly, we substitute \(z = m^2\), transforming the expression into: \[21z^2 - 32z - 5\] This makes it a standard quadratic expression in terms of \(z\). Understanding this transformation is essential for factoring quadratic-like polynomials.
quadratic formula
The quadratic formula is a crucial tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This solution method applies to any quadratic expression, whether it's expressed with variables like \(z\) or \(m^2\). In our problem, applying the quadratic formula to \[21z^2 - 32z - 5 = 0\] helps find the roots \(z = 5/21\) and \(z = -21\). Verifying these roots ensures accurate polynomial factorization, allowing back substitution to the original variable.
polynomial factorization
Polynomial factorization is the process of breaking down a polynomial into simpler parts, or 'factors,' that when multiplied together give the original polynomial.

For our example, start with identifying and rewriting the polynomial as a quadratic expression: \[21z^2 - 32z - 5\] Using the quadratic formula, we determine the roots: \(z_1 = \frac{5}{21}\) and \(z_2 = -1\). This allows us to write: \[(21z + 5)(z - 1)\]

Finally, substitute back \(z = m^2\) to get: \[(21m^2 + 5)(m^2 - 1)\]. This is further factored to \[(21m^2 + 5)(m + 1)(m-1)\], achieving complete factorization.

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Most popular questions from this chapter

Factor each trinomial. \(z^{2}(z-x)-z x(x-z)-2 x^{2}(z-x)\)

Solve each equation. (Hint: In Exercises \(51-54,\) use the substitution of variable method.) $$ 5(3 x-1)^{2}+3=-16(3 x-1) $$

Solve each equation. $$ (3 x+1)(x-3)=2+3(x+5) $$

Factor each polynomial. $$ 343 p^{3}+125 q^{3} $$

Solve each equation. $$ (5 x+1)(x+3)=-2(5 x+1) $$
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