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Chapter 8: Problem 96

Use a pattern to factor. Identify any prime polynomials. $$25 x^{2}+4 y^{2}$$

Short Answer

Expert verified
25x^2 + 4y^2 is a prime polynomial over the real numbers.

Step by step solution

01

Recognize the form

The expression given is in the form of a quadratic trinomial. Check if it matches any common factoring patterns such as the difference of squares, perfect square trinomials, or sum of squares.
02

Identify the type of binomial

The expression looks like a sum of two squares: a^2 + b^2 where a = 5x and b = 2y. Recall that the sum of squares cannot be factored over the real numbers.
03

Conclude the polynomial type

Since a^2 + b^2 cannot be factored over the reals, and doesn't fall into a factorizable special form for real numbers, consider it a prime polynomial.

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Key Concepts

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

Sum of Squares
The sum of squares refers to an expression in the form of \( a^2 + b^2 \). In our exercise, we have \( 25x^2 + 4y^2 \) which can be identified as a sum of squares because \( 25x^2 \) is \( (5x)^2 \) and \( 4y^2 \) is \( (2y)^2. \). It's important to remember that unlike the difference of squares, a sum of squares does not have a real factorization. You can't write \( (a^2 + b^2) \) as a product of polynomials with real coefficients. This makes recognizing sums of squares key in identifying expressions that cannot be factored further in the real number system.
Quadratic Trinomials
A quadratic trinomial is a polynomial of the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. In this case, \( 25x^2 + 4y^2 \) is not a standard quadratic trinomial, but you should always first check if the polynomial fits any common patterns. Typically, with quadratic trinomials, you look for ways to factor them into a product of two binomials. For example, \( x^2 + 5x + 6 \) factors into \( (x + 2)(x + 3) \). However, not all expressions will fit neatly into these common forms, and special cases like 'sum of squares' will require recognizing that they cannot be factored over the reals.
Prime Polynomials
A prime polynomial is one that cannot be factored into the product of two or more non-constant polynomials. When dealing with expressions like \( 25x^2 + 4y^2 \), recognizing the form is crucial. Since \( 25x^2 + 4y^2 \) is a sum of squares, and the sum of squares cannot be factored over the real numbers, we classify it as a prime polynomial. This means no further simplification or factorization is possible using real numbers. Identifying prime polynomials helps to know when it's time to stop looking for factors and conclude that the polynomial is in its simplest form.

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Most popular questions from this chapter

Rewrite the radical expression in exponential notation and simplify. $$\sqrt[20]{a^{5}}$$

Rewrite the radical expression in exponential notation. $$\sqrt[3]{h^{4}}$$

In a study of broiler chickens, the daily water consumption per chicken, \(y\), depended on the age of the chicken in days, \(x\). A linear model of this relationship is $$ y=\left(\frac{5.28 \frac{\mathrm{mL}}{\text { chicken }}}{\text { age in days }}\right) x \text {. } $$ Predict the amount of water needed per day by a chicken that is 20 days old.

Rewrite the radical expression in exponential notation. $$\sqrt[3]{k^{4}}$$

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