Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 4 y^{2}-16 y+15 $$

Short answer

The polynomial factors to (2y - 3)(2y - 5).

Step-by-step solution

Identify the Polynomial Type

Recognize that the given polynomial, 4 y^{2}-16 y+15, is a quadratic polynomial in the form of ax^{2} + bx + c, where a = 4, b = -16, and c = 15.

Compute the Discriminant

Calculate the discriminant using the formula ∆ = b^2 - 4ac. Substitute the values to get: ∆ = (-16)^2 - 4(4)(15) = 256 - 240 = 16.

Check the Discriminant

Since the discriminant ∆ = 16 is a perfect square, it indicates that the polynomial can be factored into real factors.

Factor the Quadratic Polynomial

Proceed to factor the quadratic polynomial. Find two numbers that multiply to ac = 4 * 15 = 60 and add up to b = -16. These numbers are -10 and -6.Rewrite the middle term using these numbers: 4y^2 - 10y - 6y + 15.Group the terms and factor by grouping: 2y(2y - 5) - 3(2y - 5). Factor out the common binomial factor: (2y - 3)(2y - 5).

Key concepts

Headline of the respective core concept

Quadratic polynomials are ones where the highest power of the variable is 2, represented generally as \( ax^2 + bx + c \). These polynomials can often be factored, but understanding how requires some knowledge of key concepts such as the quadratic formula, discriminant, and various factoring techniques.
For our specific example, the polynomial is \( 4y^2 - 16y + 15 \), which we know is quadratic because the highest power of y is 2.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. The formula is:
\(x = \frac{-b \plusmn \sqrt{b^2 - 4ac}}{2a}\).
This formula gives solutions for any quadratic equation of the form \(ax^2 + bx + c = 0\). Here’s how it works:

• You plug in the coefficients a, b, and c from your polynomial.
• Calculate the discriminant (more on this later).
• Solve for x using simple addition, subtraction, multiplication, and division.

In our case, \(a = 4, b = -16, c = 15\). Although we didn't use the quadratic formula directly to factor the polynomial here, understanding it aids in grasping the overall factorization process, especially when dealing with more complex scenarios.

Discriminant

The discriminant is a part of the quadratic formula and is calculated as \(\Delta = b^2 - 4ac\). This value helps to determine the nature of the roots for a quadratic polynomial:

• If \(\Delta > 0\), there are two distinct real roots, meaning the polynomial can be factored.
• If \(\Delta = 0\), there is exactly one real root, meaning the polynomial can still be factored but only one solution exists.
• If \(\Delta < 0\), there are no real roots, meaning the polynomial cannot be factored using real numbers.

In our example, \(\Delta = (-16)^2 - 4(4)(15) = 256 - 240 = 16\). Since 16 is a perfect square and greater than zero, we know the quadratic polynomial can be factored.

Factoring Techniques

Factoring a quadratic polynomial involves breaking it into products of two binomials. Here’s how we did it with \(4y^2 - 16y + 15\):
- First, identify two numbers that multiply to \(ac = 4 \times 15 = 60\) and add up to b = -16.
- These numbers are -10 and -6.
- Rewrite the middle term \( -16y \) as \( -10y - 6y \).
- Express the polynomial: \(4y^2 - 10y - 6y + 15\).
- Factor by grouping: \(2y(2y - 5) - 3(2y - 5)\).
- Factor out the common binomial: \( (2y - 5)(2y - 3)\).

This method is incredibly useful as it allows breaking down more complex expressions into simpler factors.