Problem 37 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 3 d^{2}+3 d-5 $$

Step-by-Step Solution

Answer

The polynomial 3d^2 + 3d - 5 is a prime polynomial.

Step 1: Identify and write down the polynomial

The given polynomial is: \[3d^2 + 3d - 5\].

Step 2: Calculate the discriminant

Use the discriminant formula for a quadratic polynomial, which is given by: \[ \text{Discriminant} = b^2 - 4ac \]. For the polynomial \[3d^2 + 3d - 5\], the coefficients are: \( a = 3 \), \( b = 3 \), and \( c = -5 \). Calculating the discriminant: \[ 3^2 - 4 \times 3 \times (-5) = 9 + 60 = 69 \].

Step 3: Analyze the discriminant

Since the discriminant \(69\) is not a perfect square, the quadratic does not factorize nicely with integer coefficients.

Step 4: Conclusion on factorization

Given that the discriminant is not a perfect square, it can be concluded that the polynomial \[3d^2 + 3d - 5\] does not factorize further and is therefore a prime polynomial.

Key Concepts

discriminant

The discriminant is a special number that you can calculate from the coefficients of a quadratic polynomial to determine several key properties of the polynomial. The formula for the discriminant is given by: discriminant = b^2 - 4ac To put it simply, the discriminant helps us decide if a quadratic equation can be factorized into real numbers (numbers you can plot on a regular number line). Let's look at the polynomial 3d^2 + 3d - 5. to understand the discriminant's role better. Here, a is 3, b is 3, and c is -5. When we plug these into the formula, we get: discriminant = 3^2 - 4 ⋅ 3 ⋅ -5 = 9 + 60 = 69. Since 69 is not a perfect square, the quadratic polynomial does not factorize into nice integers. This leads us directly to the prime polynomial concept, because the discriminant tells us a lot about how the polynomial behaves.

When the discriminant is a

• positive perfect square, the polynomial factorizes nicely,
• positive but not a perfect square, factorizing into irrational numbers,
• zero, it simplifies to a squared binomial,
• negative, it has no real roots but complex ones instead.

quadratic polynomial

A quadratic polynomial is one where the highest power of the variable (often labeled x or d) is 2. The standard form is written as ax^2 + bx + c, where a, b, and c are coefficients. You encounter quadratic polynomials in many areas of math, including algebra, geometry, and calculus.

For the quadratic polynomial 3d^2 + 3d - 5, here's the breakdown:

• a = 3 (the coefficient in front of d^2),
• b = 3 (the coefficient in front of d),
• c = -5 (the constant term).

Quadratic polynomials can be factored or not, depending on the discriminant, as we've seen earlier. They have various applications such as solving quadratic equations, formulating mathematical models, and in physics problems involving motion. Understanding how to manipulate and solve quadratic equations is a foundational skill in higher-level math.

prime polynomial

A prime polynomial is much like a prime number in arithmetic. It cannot be factored into the product of two lower-degree polynomials with integer coefficients. In the case of integer coefficients, a prime polynomial does not break down into simpler polynomials that multiply to give the original polynomial.

For 3d^2 + 3d - 5, the discriminant calculation showed that 69 is not a perfect square. This means we can't factor the polynomial into simpler factors with integer coefficients. Therefore, the polynomial is considered a prime polynomial.

Understanding prime polynomials helps you identify when you can no longer break down an equation, simplifying the problem-solving process. When dealing with prime polynomials, remember:

• The discriminant helps decide factorability,
• If factorable, the breakdown will give us simpler binomials,
• If not factorable (like 3d^2 + 3d - 5), it remains prime.

Knowing when a polynomial is prime versus when it can be factored is crucial in simplifying and solving mathematical problems.