Problem 79 Create a trinomial of the form \... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 6: Problem 79

Create a trinomial of the form $a x_{2}+b x+c$ that does not factor and share it along with the reason why it does not factor.

Short Answer

Expert verified

The trinomial $5x^2 - 6x + 2$ does not factor because its discriminant $(-4)$ is negative.

Step by step solution

Choose Values for a, b, and c

Select integer values for $a$, $b$, and $c$ such that the trinomial does not factor neatly over the integers. Let's choose $a = 5$, $b = -6$, and $c = 2$. So, the trinomial is $5x^2 - 6x + 2$.

Confirm Non-factorability Using the Discriminant

For a trinomial $ax^2 + bx + c$, check if it can be factored over the integers by using the discriminant: $b^2 - 4ac$. Substitute $a = 5$, $b = -6$, and $c = 2$: $(-6)^2 - 4 \times 5 \times 2 = 36 - 40 = -4$.

Interpret the Discriminant Result

Since the discriminant $(-4)$ is negative, there are no real roots. Hence, the trinomial does not factor over the integers.

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Key Concepts

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

Discriminant Method

When dealing with trinomials, one effective way to determine if a trinomial is factorable is by using the discriminant method. This involves the formula:

For a quadratic equation of the form $ ax^2 + bx + c $, the discriminant $ \Delta $ is defined as $ b^2 - 4ac $.

The discriminant can tell us about the nature of the roots of the equation. Here's how you can interpret it:

If $ \Delta > 0 $, the equation has two distinct real roots, indicating that it can be factored over the real numbers.
If $ \Delta = 0 $, there is exactly one real root (a repeated root), and the trinomial can still be factored as a perfect square.
If $ \Delta < 0 $, the equation has no real roots, which means it cannot be factored over the integers.

For example, in the trinomial $ 5x^2 - 6x + 2 $, we calculated the discriminant to be $-4$. This negative discriminant confirms that no real roots exist, hence the trinomial cannot be neatly factored over the integers.

Integer Coefficients

In the realm of polynomial equations, the coefficients play a significant role. For a polynomial to be considered over the integers, all of its coefficients $ a $, $ b $, and $ c $ must be integers themselves. When performing operations like factoring, it's crucial to ensure these coefficients remain integers if we want the factorization to be valid over the set of integers.

Integer coefficients simplify arithmetic operations like addition, subtraction, and multiplication, as these operations result in integer outputs.
Within the context of factorization, a common goal is to express a polynomial as a product of other polynomials with integer coefficients.

By selecting integers like $ a = 5 $, $ b = -6 $, and $ c = 2 $ for the trinomial $ 5x^2 - 6x + 2 $, we ensure every step of checking factorability aligns with integer arithmetic, a fundamental aspect when dealing with such problems.

Non-factorable Polynomials

Non-factorable polynomials, sometimes known as irreducible polynomials, are those that cannot be expressed as a product of two or more non-trivial polynomials with integer coefficients. This property means that within the context of integer factorization, the polynomial is "prime."

For a quadratic like $ ax^2 + bx + c $, if the discriminant $ b^2 - 4ac $ is less than zero, it suggests real solutions are nonexistent, making the trinomial non-factorable over the integers.
Non-factorable polynomials pose challenges for solving equations as traditional factorization techniques do not apply. Instead, numeric or more advanced algebraic methods must be employed.

Take $ 5x^2 - 6x + 2 $ as a classic instance. Here, the discriminant is $-4$, which leads to complex roots, confirming our trinomial remains non-factorable over the integers. Understanding this property helps significantly in algebraic problem-solving, as it delineates the limits of simpler solution methods.

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91影视

Create a trinomial of the form \(a x_{2}+b x+c\) that does not factor and share it along with the reason why it does not factor.

Short Answer

Step by step solution

Choose Values for a, b, and c

Confirm Non-factorability Using the Discriminant

Interpret the Discriminant Result

Key Concepts

Discriminant Method

Integer Coefficients

Non-factorable Polynomials

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