Textbook Question
Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.
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0. Review of Algebra
Factoring Polynomials
1:53 minutesProblem 133
Textbook Question
Factor each polynomial over the set of rational number coefficients. 49x2-1/25
Verified step by step guidance1
Recognize that the polynomial \(49x^2 - \frac{1}{25}\) is a difference of squares because it can be written as \((7x)^2 - \left(\frac{1}{5}\right)^2\).
Recall the difference of squares factoring formula: \(a^2 - b^2 = (a - b)(a + b)\).
Identify \(a = 7x\) and \(b = \frac{1}{5}\) in the expression.
Apply the formula to factor the polynomial as \((7x - \frac{1}{5})(7x + \frac{1}{5})\).
Write the final factored form clearly, ensuring all terms are expressed with rational coefficients.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of simpler polynomials. This process simplifies expressions and solves equations. Recognizing special forms like difference of squares helps in factoring efficiently.
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Difference of Squares
The difference of squares is a specific factoring pattern where an expression of the form a² - b² can be factored as (a - b)(a + b). This is useful for polynomials like 49x² - 1/25, where both terms are perfect squares.
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Rational Number Coefficients
When factoring over rational numbers, all coefficients in the factors must be rational numbers (fractions or integers). This restricts the factorization to expressions with rational coefficients, excluding irrational or complex numbers.
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