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Chapter 5: Problem 57

Factor completely. See Examples 1 through \(9 .\) $$ -16 y^{2}+64 $$

Short Answer

Expert verified
The factored form is \\(-16(y-2)(y+2)\\).

Step by step solution

01

Identify Common Factors

Firstly, examine the terms \(-16y^2\) and \(+64\). Notice that both terms have \(-16\) as a common factor. Factor out \(-16\) from both terms.This gives:\[-16(y^2 - 4)\]
02

Recognize and Apply Difference of Squares

The expression inside the parentheses, \((y^2 - 4)\), is a difference of squares since it can be written as \(y^2 - 2^2\). Recall that a difference of squares can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\).Apply this to the expression:\[y^2 - 4 = (y - 2)(y + 2)\]
03

Write the Final Factored Form

Substitute the factored form of \(y^2 - 4\) back into the expression we factored out in Step 1. Thus, the expression becomes:\[-16(y - 2)(y + 2)\]
04

Verify Your Solution

Expand \(-16(y-2)(y+2)\) to check the factoring:\[-16((y-2)(y+2)) = -16(y^2 - 4)\]This simplifies to \(-16y^2 + 64\), which matches the original expression, confirming the solution is correct.

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Key Concepts

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

Difference of Squares
The concept of a difference of squares is a key tool in factoring. It involves two terms, each of which is a perfect square, separated by a subtraction sign. Any expression of the form \(a^2 - b^2\) can be factored into \((a - b)(a + b)\). This is because \((a - b)(a + b) = a^2 - b^2\), based on the expansion of the binomials.
Understanding this identity allows you to quickly break down expressions that fit this pattern. For example, in the stepped solution, \(y^2 - 4\) is a difference of squares:
  • \(y^2\) is \((y)^2\)
  • 4 is \(2^2\)
  • Hence, \(y^2 - 4 = (y - 2)(y + 2)\)
This factoring method is particularly straightforward when you recognize that the terms are perfect squares and are subtracted from each other.
Common Factor
A common factor is a shared divisor among all terms in an expression. Recognizing and factoring out a common factor is a crucial step in simplifying polynomials. For example, in the given expression \(-16y^2 + 64\), both terms, \(-16y^2\) and \(64\), have \(-16\) as a common factor:
  • Divide each term by \(-16\)
  • Extract this factor: \(-16(y^2 - 4)\)
By factoring out the greatest common factor, you simplify the expression, making it easier to recognize other factoring opportunities, like the difference of squares. This step cleans up the polynomial, revealing simpler expressions ready for further factoring.
Polynomial Factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors. It's like breaking down a number into its prime factors, but with algebraic expressions. In our step-by-step solution,
the original polynomial \(-16y^2 + 64\) is effectively factored by performing these steps:
  • Identify a common factor: \(-16\)
  • Remaining expression: \(y^2 - 4\)
  • Recognize \(y^2 - 4\) as a difference of squares
  • Factor it as \((y - 2)(y + 2)\)
  • Finally, combine all factors to get \(-16(y-2)(y+2)\)
The ultimate goal of polynomial factorization is to express the polynomial in a more simplified, multiplied form that can be used for solving equations, graphing, or further simplifications. It's an essential skill that aids in understanding the structure of polynomials.

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Most popular questions from this chapter

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-36 $$

Which numbers are equal to \(36,000 ?\) Of these, which is written in scientific notation? $$a. 36 \times 10^{3}$$ $$b.360 \times 10^{2}$$ $$c. 0.36 \times 10^{5}$$ $$d. 3.6 \times 10^{4}$$

Multiply. See Section 5.4. $$ (3 x+5)^{2} $$

Factor each polynomial completely. See Examples 1 through 12. $$ 3 x^{2}+6 x-45 $$

Multiply. See Section 5.4. $$ (x-2)\left(x^{2}+2 x+4\right) $$
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