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Chapter 0: Problem 106

Factor completely. $$ 7 x^{4}+34 x^{2}-5 $$

Short Answer

Expert verified
The factored form of the expression \(7 x^{4}+34 x^{2}-5\) is \(7(x^{2} - 1)(x^{2} + 5)\).

Step by step solution

01

Rewrite the Polynomial

Rewrite the given polynomial treating \(x^{2}\) as the variable: \(7 (x^{2})^{2} + 34 x^{2} - 5\). This polynomial will be treated as having the form \(ax^{2} + bx + c\).
02

Apply the Quadratic Formula

To solve for \(x^{2}\), the quadratic formula end up being used: \(x^{2} = \frac{-b \pm \sqrt{b^{2} - 4ac} }{2a}\). The values of \(a\), \(b\), and \(c\) in this context are 7, 34, and -5 respectively.
03

Compute the Discriminant

The discriminant, \(b^{2} - 4ac\), is computed first. Substituting the values, it becomes \(34^{2} - 4(7)(-5) = 1156 + 140 = 1296\).
04

Solve for \(x^{2}\)

We substitute all the values to the quadratic equation: \(x^2 = \frac{-34 \pm \sqrt{1296}}{14}\) which simplifies to \( x^{2} = 1, -5 \). Thus, it gives us two results.
05

Factoring the Result

The results are the roots for the quadratic in terms of \(x^{2}\). Hence, the factored form of the expression is \(7(x^{2} - 1)(x^{2} + 5)\).

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool used to find the solutions, or roots, of quadratic equations. A quadratic equation is typically in the form of: \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps to directly calculate the values of \(x\) for any quadratic equation.
Key aspects include:
  • The term \(\sqrt{b^2 - 4ac}\) is known as the discriminant.
  • The formula provides up to two solutions, corresponding to the \(\pm\) symbol in \(\pm \sqrt{b^2 - 4ac}\).
  • It is applicable when factoring the quadratic manually is complex or not readily evident.
Discriminant
The discriminant in quadratic equations plays a crucial role in determining the nature of the roots. It is found under the square root symbol within the quadratic formula and is given by:\[ b^2 - 4ac \]
The value of the discriminant can indicate:
  • Positive: Two distinct real roots. For instance, in the given solution, the discriminant is 1296, a perfect square, leading to real solutions.
  • Zero: One real root (or a repeated root). The solution to the equation in this case is a perfect square.
  • Negative: No real roots, but two complex conjugates. The roots do not intersect the x-axis in the graph.
Factoring Polynomials
Factoring polynomials involves breaking down a complex polynomial expression into simpler, multiplied factors. This process reveals the roots or solutions in an alternative and sometimes more intuitive manner.
For the expression \(7x^4 + 34x^2 - 5\), we first rewrite it in a quadratic form in terms of \(x^2\) and use the quadratic formula:
  • Solve \(7(x^2)^2 + 34x^2 - 5\) to find \(x^2 = 1\) and \(x^2 = -5\).
  • Express as \(7(x^2 - 1)(x^2 + 5)\), showing each factor leading to zero will help identify the roots.
The factors indicate that setting each factor to zero will provide the original polynomial's solutions.
Roots of a Polynomial
Roots of a polynomial are the values of the variable that satisfy the polynomial equation (i.e., make it equal to zero). These roots are vital for understanding the behavior and graph of a polynomial.
In the given task, by applying the quadratic formula, the expression \(7(x^2 - 1)(x^2 + 5)\) gives us the roots when each factor is set to zero:
  • From \(x^2 - 1 = 0\), we get real roots \(x = \pm 1\).
  • From \(x^2 + 5 = 0\), we get no real roots but complex roots \(x = \pm i\sqrt{5}\).
Comprehending polynomial roots helps in many endeavors, such as graphing, optimization, and solving real-world problems.

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