Problem 43 Factor completely. If a polynomi... [FREE SOLUTION]

Chapter 6: Problem 43

Factor completely. If a polynomial is prime, state this. $$ 12 a^{2}-17 a b+6 b^{2} $$

Short Answer

Expert verified

(n - 5 + 3m)(n - 5 - 3m)

Step by step solution

Identify possible factorizations

First, notice that the polynomial can be rewritten as the difference of two squares. We have the expression: n^2 - 10n + 25 - 9m^2

Recognize perfect squares

Observe that both n^2 - 10n + 25 and 9m^2 are perfect squares. Specifically: n^2 - 10n + 25 = (n - 5)^2 9m^2 = (3m)^2

Rewrite as difference of squares

Rewrite the polynomial in the form of the difference of squares: (n - 5)^2 - (3m)^2

Apply the difference of squares formula

Use the formula for the difference of squares, a^2 - b^2 = (a + b)(a - b), where a = n - 5 and b = 3m: (n - 5 + 3m)(n - 5 - 3m)

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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 notable concept in algebra. It refers to an expression in the format: \[a^2 - b^2\]. \[a^2 - b^2\] can be factored into \[(a + b)(a - b)\]. Meaning, the difference between two squared terms equals the product of their sum and their difference. For example, if we have \[x^2 - 16\], we can express this as \[x^2 - 4^2\], which factors to \[(x + 4)(x - 4)\]. Recognizing this pattern makes factoring such expressions straightforward and is key to breaking down complex polynomial expressions.

Perfect Squares

Let's talk about 'perfect squares'. A perfect square is a number or expression that can be written as a square of another term. For example, 36 is a perfect square since it is $6^2$. In algebra, expressions like $ (a + b)^2 $ or $ (a - b)^2 $ are perfect squares. Recognizing perfect squares helps in simplifying and factoring polynomials. For instance, $ n^2 - 10n + 25 $ is a perfect square because it can be written as $ (n-5)^2 $. Spotting these patterns enables us to rewrite expressions neatly and factor them efficiently.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler 'factor' polynomials. These factor polynomials, when multiplied together, give back the original polynomial. There are several methods to factor polynomials, such as factoring out the greatest common factor (GCF), using the difference of squares, and recognizing perfect squares. In our example, we have $ n^2 - 10n + 25 - 9m^2 $. This gets simplified by recognizing that $ (n - 5)^2 $ and $ (3m)^2 $ are perfect squares. This allows us to apply the difference of squares formula: \[ (n-5)^2 - (3m)^2 = (n-5 + 3m)(n-5 - 3m) \]. Factoring polynomials helps simplify expressions and solve polynomial equations.

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91影视

Factor completely. If a polynomial is prime, state this. $$ 12 a^{2}-17 a b+6 b^{2} $$

Short Answer

Step by step solution

Identify possible factorizations

Recognize perfect squares

Rewrite as difference of squares

Apply the difference of squares formula

Key Concepts

Difference of Squares

Perfect Squares

Factoring Polynomials

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