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Chapter 0: Problem 86

Factor completely, or state that the polynomial is prime. $$x^{2}-10 x+25-36 y^{2}$$

Short Answer

Expert verified
The completely factored form of the polynomial \(x^{2}-10 x+25-36 y^{2}\) is: (x - 5 + 6y)(x - 5 - 6y)

Step by step solution

01

Identify Perfect Squares

The given equation can be written as:\n \(x^2 - 2*5*x + (5^2) - (6y)^2\).\nFrom this, it can be seen that both \(x^2 - 2*5*x + (5^2)\) and \((6y)^2\) are perfect squares.
02

Factor Perfect Squares

\(x^2 - 2*5*x + (5^2)\) is a perfect square and can be written as \((x - 5)^2\), while \((6y)^2\) can be written as \((6y)^2\). So the expression can be rewritten as \((x - 5)^2 - (6y)^2\).
03

Factor Using Difference of Squares Method

Since \((x - 5)^2 - (6y)^2\) is a difference of two squares, use the difference of squares formula \((a^2 - b^2) = (a + b)(a - b)\) to factorize the expression. Here, \(a = x - 5\) and \(b = 6y\). Therefore, \((x - 5)^2 - (6y)^2 = (x - 5 + 6y)(x - 5 - 6y)\).

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Key Concepts

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

Difference of Squares
The Difference of Squares is a special algebraic identity that makes factoring easier when dealing with two perfect squares separated by a minus sign. This expression takes the form \(a^2 - b^2\). When factored, it becomes \((a + b)(a - b)\). In many algebra problems, recognizing this pattern can simplify the solution significantly.
  • Each component \(a\) and \(b\) represents a squared term.
  • To use this method, identify two terms which are perfect squares.
  • They must be subtracted from each other; hence 'Difference' of Squares.
For example, consider \((x - 5)^2 - (6y)^2\). Here, \(a\) is \(x - 5\), and \(b\) is \(6y\). When evaluated using the formula, it results in \((x - 5 + 6y)(x - 5 - 6y)\). This elegant identity is an essential tool for efficiently cracking polynomial expressions.
Perfect Squares
Perfect Squares are terms that result from squaring a binomial expression. For instance, \(x^2 - 2 \cdot 5 \cdot x + 5^2\) is a perfect square. This comes from \((x - 5)^2\), indicating it originated from the square of the binomial \(x - 5\). This makes the expression very straightforward to factor, because it simply undoes the squaring.
  • A perfect square trinomial often looks like \(a^2 + 2ab + b^2\), where it can be rewritten as \((a + b)^2\).
  • For subtraction-based trinomials, you would look for patterns like \(a^2 - 2ab + b^2\) to rewrite as \((a - b)^2\).
Understanding these results makes problems involving polynomials much easier to factor, because once identified, they can be reduced quickly to their binomial origins.
Polynomial Expressions
Polynomial Expressions form the foundation for many algebraic operations. These expressions can consist of variables, coefficients, addition, subtraction, and multiplication, all raised to non-negative integer powers. In mathematics, dealing with polynomial expressions involves simplifying or factoring them to find base elements or roots.
  • Polynomial terms are typically classified by degree鈥攖he highest power present in the expression.
  • Factoring a polynomial is breaking it down into the simplest terms called factors, which can be multiplied to give the original polynomial.
  • For example, factoring \(x^2 - 10x + 25 - 36y^2\) can involve recognizing perfect squares and then using identities like the Difference of Squares.
Approaching polynomial expressions with these strategies aids in solving, simplifying, and even graphing polynomial functions in both education and real-world applications.

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Most popular questions from this chapter

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \cdot \frac{8}{15}$$

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

The formula for converting Fahrenheit temperature, \(F,\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)

Solve each equation. $$\left|x^{2}+2 x-36\right|=12$$
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