Problem 16 - Elementary Algebra

Question

Use the guess and check method to factor. Identify any prime polynomials. $$ y^{2}+16 y-64 $$

Step-by-Step Solution

Answer

The polynomial \[ y^2 + 16y - 64 \] is a prime polynomial.

Step 1 - Understanding the problem

Recognize that the given polynomial is in the form of a quadratic equation: \[ y^2 + 16y - 64 \]

Step 2 - Setting up the guess and check method

We are looking for two binomials (\[(y + a)(y + b)\]) such that when multiplied, they give us the original polynomial. This means: \[ y^2 + (a+b)y + ab = y^2 + 16y - 64 \]

Step 3 - Find the product and sum

The sum \[a + b\] should be equal to 16, and the product \[ab\] should be equal to -64. We need to find integer pairs that satisfy these conditions.

Step 4 - Testing pairs

Try possible pairs for a and b: 1. \[ (20) + (-4) = 16 \] and \[ (20) \times (-4) = -80 \]2. \[ (-8) + (8) = 0 \] and \[ (-8) \times (8) = -64 \]3. \[ (18) + (-2) = 16 \] and \[ (18) \times (-2) = -36 \]None of these pairs work.

Step 5 - Conclusion

Since no pairs satisfy both the sum and product conditions, the polynomial \[ y^2 + 16y - 64 \] is a prime polynomial.

Key Concepts

guess and check method

The guess and check method is a technique used to factor polynomials. This method involves making educated guesses for the factors of the polynomial, and then checking to see if those factors multiply back to the original polynomial.
For example, with the polynomial \(y^2 + 16y - 64\), you would guess possible pairs of numbers \(a\) and \(b\) that could multiply back to this polynomial when placed in the binomial form \( (y + a)(y + b) \).
1. Start by determining the relationships between the factors:

• The sum of \(a\) and \(b\) should be equal to the coefficient of the middle term (16 in this case).
• The product of \(a\) and \(b\) should be equal to the constant term (−64 in this case).

The process can be repetitive, as it requires trying several pairs of integers to see if they work. This method may seem tedious, but it is often effective in finding the right factors for simpler quadratic equations.

quadratic equations

A quadratic equation is a type of polynomial equation of the form \[ax^2 + bx + c = 0\]. In our exercise, we are dealing with \[y^2 + 16y - 64\], which is in standard quadratic form with \(a = 1\), \(b = 16\), and \(c = -64\).
Quadratic equations are frequently encountered in algebra, and they can often be solved by factoring, which is the goal here.
Understanding quadratic equations involves recognizing their standard structure:

• They are characterized by having the highest power of the variable (y in this case) be 2.
• They can often be factored into the product of two binomial expressions.

Factoring involves finding two binomials that multiply to give the original quadratic expression. When efforts to find such factors fail, as in this exercise, it may indicate that the polynomial is prime.

prime polynomials

A polynomial is considered prime when it cannot be factored into the product of two non-trivial polynomials with integer coefficients. In simpler terms, it means there are no two polynomials that multiply to form the original polynomial.
For the polynomial \[y^2 + 16y - 64\], after exhausting possible pairs for \(a \) and \(b\) using the guess and check method, none of them satisfy both the required sum and product conditions.
This indicates that \[y^2 + 16y - 64\] cannot be factored into simpler polynomials with integer coefficients, making it a prime polynomial.
Identifying prime polynomials can be important in higher-level algebra, as it simplifies many problems by confirming that no further factorization is possible.

integer pairs

In factoring, especially using the guess and check method, finding appropriate integer pairs that satisfy given conditions is crucial.
For the quadratic polynomial \[y^2 + 16y - 64\], we need to find pairs of integers (a, b) such that:

• The sum \((a + b)\) equals the coefficient of the middle term, which is 16.
• The product \((ab)\) equals the constant term, which is -64.

Testing pairs involves checking several combinations:
1. \((20, -4)\) does not work because \(20 \times -4 = -80\), not -64.
2. \((-8, 8)\) does not work because \(-8 + 8 = 0\), not 16.
3. \((18, -2)\) does not work because \(18 \times -2 = -36\), not -64.
As seen here, none of these pairs work, leading to the conclusion that the polynomial is prime.
Understanding how to find and test integer pairs effectively can help tackle many factoring problems in algebra.