Problem 48 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ y^{2}-400 $$

Step-by-Step Solution

Answer

(y-20)(y+20) is the factored form, and both factors are prime polynomials.

Step 1 - Identify the form of the polynomial

The given polynomial is in the form of a difference of squares, which can generally be written as a^{2}-b^{2}. Notice that y^{2} is a perfect square and 400 is also a perfect square because it can be written as 20^{2}

Step 2 - Apply the difference of squares formula

Use the difference of squares formula: a^{2}-b^{2} = (a-b)(a+b). In this case: a=y and b=20 . This gives us: y^{2}-400=(y-20)(y+20)

Step 3 - Verify the factorization

Multiply the factors to check the factorization: (y-20)(y+20) = y^{2}+20y-20y-400 n = y^{2} -400 n . The factorization is correct as it simplifies back to the original expression.

Step 4 - Identify prime polynomials

Check if the factors (y-20) and (y+20) can be factored further. Since they have no common factors other than 1 and are not further factorizable polynomials, they are prime coefficients.

Key Concepts

difference of squares

The concept of the difference of squares is central in algebra. It appears frequently and makes many problems easier to solve.
When you see a polynomial written as \(a^2 - b^2\), you can recognize it as a difference of squares.
Here, \(a\) and \(b\) are perfect squares. A perfect square is any number that can be expressed as another number squared.
For the polynomial \( y^{2} - 400\), we can rewrite it as \((y)^2 - (20)^2\).
This matches the form \(a^2 - b^2\), where \(a = y\) and \(b = 20\).
The formula for factoring a difference of squares is: \(a^2 - b^2 = (a - b)(a + b)\).
Applying this to our example, we get: \( y^{2}-400=(y-20)(y+20) \).
This simplification helps break down complex problems into smaller, more manageable factors.

prime polynomials

Prime polynomials cannot be factored any further using integer coefficients.
Just like prime numbers, which are only divisible by 1 and themselves, prime polynomials are indivisible.
In the polynomial \( y^{2} - 400 \), we factored it into \( (y - 20)(y + 20) \).
Next, we check if these factors are prime.
Both \(y - 20\) and \(y + 20\) have no common factors other than 1.
Therefore, we cannot break them down any further. This indicates that \(y - 20\) and \(y + 20\) are prime polynomials.
Recognizing prime polynomials helps in simplifying and solving algebraic expressions.

perfect square

A perfect square is a number that is the square of an integer.
When factoring polynomials, identifying perfect squares simplifies the problem.
For example, in \( y^{2} - 400 \), both \( y^2 \) and \( 400 \) are perfect squares.
\( y^2 \) can be written as \( y \times y \), and \( 400 \) as \( 20 \times 20 \).
This identification is crucial because it allows the use of the difference of squares formula.
Recognizing perfect squares quickly and accurately makes factoring more intuitive and straightforward.
Remember: constant practice and thorough understanding of perfect squares will enhance your problem-solving skills in algebra.