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Chapter 6: Problem 19

Factor completely. If a polynomial is prime, state this. $$ 8 x^{2}-16-28 x $$

Short Answer

Expert verified
The polynomial \(t^2 + 25\) is prime and cannot be factored.

Step by step solution

01

Identify the Type of Polynomial

The given polynomial is a quadratic expression, specifically a sum of squares: \(t^2 + 25\).
02

Check for Factoring Patterns

Recall common factoring patterns such as the difference of squares \(a^2 - b^2 = (a - b)(a + b)\). However, there is no direct factoring pattern for the sum of squares \(a^2 + b^2\).
03

Verify for Prime

Since \(t^2 + 25\) does not meet any known factoring patterns and cannot be factored into real-number polynomials, it is considered a prime polynomial over the real numbers.

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Key Concepts

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

Quadratic Expression
When dealing with a quadratic expression, you're looking at a polynomial of the form \( ax^2 + bx + c \). Here, the coefficient \( a \) is attached to the square term, \( b \) to the linear term, and \( c \) is the constant.

Quadratics are very common in algebra and come in various forms. When simplified, they primarily look like:
  • \( ax^2 + bx + c \)
  • \( ax^2 + c \) when there is no linear term
  • \( ax^2 \) when the polynomial is just a perfect square

In the given polynomial \( t^2 + 25 \), it is simplified to just the sum of a square and a constant. This makes it a special type of quadratic expression, specifically a sum of squares. Understanding the basic structure of quadratic polynomials helps in identifying their types and potential factoring methods.
Sum of Squares
A sum of squares occurs when you have a polynomial in the form \( a^2 + b^2 \). Unlike the difference of squares, which can be factored as \( (a - b)(a + b) \), the sum of squares doesn't have a straightforward factorization pattern involving real numbers.

For instance, in the example \( t^2 + 25 \), we see:
  • \( t^2 = t \cdot t \)
  • 25 is a constant, which can be written as \( 5^2 \)

Despite identifying the individual squares, the sum \( t^2 + 25 \) does not fit into patterns like the difference of squares or perfect square trinomials. Because of this, it is left in its summed square state. Understanding it as a sum of squares helps clarify why it can鈥檛 be factored further using standard algebraic techniques.
Prime Polynomial
A polynomial is considered prime when it cannot be factored into the product of two or more non-constant polynomials. Think of prime polynomials as the prime numbers of the polynomial world - they can't be broken down any further over the set of real numbers.

For the polynomial \( t^2 + 25 \), this piece can't be separated into simpler polynomials using real coefficients.
  • It doesn't fit any known patterns like difference of squares or trinomials.
  • There are no factors that evenly divide into it.

Consequently, the polynomial \( t^2 + 25 \) is prime. Understanding prime polynomials is important for determining when you've simplified an expression as much as possible.

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Most popular questions from this chapter

Solve. Round any irrational solutions to the nearest thousandth. $$ r^{2}+16=8 r $$

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{4 a}-49 y^{2 a} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ q^{2}+1 $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}=\frac{1}{100} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 2 a^{4}-32 y^{8} $$
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