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Chapter 0: Problem 85

Factor completely, or state that the polynomial is prime. $$x^{2}-12 x+36-49 y^{2}$$

Short Answer

Expert verified
The completely factored form of the given polynomial is \((x-6 -7y) (x-6 + 7y)\)

Step by step solution

01

Identify the pattern

In the given expression \(x^{2}-12 x+36-49 y^{2}\), the expressions \(x^{2}-12 x+36\) and \(49y^{2}\) can be rewritten as \( (x-6)^2 \) and \((7y)^2\) respectively. This brings it to the format \(a^2 - b^2\).
02

Apply the difference of squares method

The difference of squares method is applied to factorize the polynomial. This method is based on the formula \( a^2 鈥� b^2 = (a + b)(a - b)\). Here, \(a = (x-6)\) and \(b = (7y)\). So, the polynomial decomposed into \((x-6 -7y) (x-6 + 7y)\).

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=|x+1|,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -4 and ending with 2

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.
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