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Chapter 6: Problem 52

Factor completely. If a polynomial is prime, state this. $$ 7 a^{2} b^{2}+13 a b+6 $$

Short Answer

Expert verified
The factored form is \[ (5z + y)^2 \].

Step by step solution

01

- Recognize the Quadratic Form

Observe that the polynomial given is a quadratic in the form of \[ ax^2 + bxy + cy^2 \]. Identify the coefficients a, b, and c for:\[ 25z^2 + 10zy + y^2 \].
02

- Identify the Perfect Squares

Check if the terms are perfect squares. Notice that:\[ 25z^2 = (5z)^2 \]\[ y^2 = y^2 \]
03

- Check the Middle Term

Determine if the middle term, 10zy, fits the form \[ 2(\text{first term})(\text{second term}) \]. Here, \[ 2(5z)(y) = 10zy \], which matches the middle term.
04

- Write the Factored Form

Since the polynomial fits the form of \[ (ax + by)^2 \], rewrite it as a binomial squared:\[ (5z + y)^2 \]
05

- Verify the Result

Expand \[ (5z + y)^2 \] to make sure it equals the original polynomial:\[ (5z + y)(5z + y) = 25z^2 + 10zy + y^2 \]. The factorization is correct.

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Key Concepts

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

quadratic form
The quadratic form is central in algebra and polynomial factoring. Recognizing it makes the factorization process simpler. A polynomial is in quadratic form when it matches o line i.e., ax^2 + bxy + cy^2. In our given exercise, we identify the coefficients for the quadratic polynomial 25z^2 + 10zy + y^2 as:
  • a = 25
  • b = 10
  • c = 1
. Recognizing this form helps us understand the relationships between the terms and guides us towards factorization. Make sure you identify each term correctly so the rest of the steps go smoothly.
perfect squares
Perfect squares help simplify the process of determining factors of a polynomial. A perfect square is a number or expression that can be expressed as some integer or algebraic term squared. For our polynomial given, we identify:
  • 25z^2 is a perfect square, as it can be written as (5z)^2
  • y^2 is also a perfect square, since it is straightforwardly expressed as y^2
. Identifying these squares tells us the polynomial has a structure that can be broken down into simpler, multiplied forms. Effort observing perfect squares in your polynomial allows quick recognition of binomial parts.
binomial squared
A binomial squared is an expression of the form (ax + by)^2 . It results from multiplying a binomial by itself. For example, the expression can be expanded as: the binomial (5z + y) squared: (you have (5z + y)(5z + y)). The expansion gives: <=> 25z^2 + 10zy + y^2, exactly matching our original polynomial. Recognizing binomial squared can help factorize complex polynomials swiftly, turning the problem into read the factoring a simple binomial and squaring it. Always verify your factorization by expanding back to check if it matches the original polynomial. Wanting more such intuitive findings solidifies your understanding of polynomials' behavior and structure.

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Most popular questions from this chapter

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{2}-\left(\frac{1}{x}\right)^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 x^{4}-25 x^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9-x^{2}-2 x y-y^{2} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}-9=0 $$

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$
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