Problem 24 - Elementary Algebra

Question

Use the \(a c\) method to factor. Check the factoring. Identify any prime polynomials. $$ 2 x^{2}+9 x-35 $$

Step-by-Step Solution

Answer

The factors are \[(2x - 5)(x + 7)\].

Step 1: Multiply a and c

In the quadratic expression \[2x^2 + 9x - 35\], identify the coefficients a, b, and c where a = 2, b = 9, and c = -35. Multiply a and c: \[2 \times -35 = -70\].

Step 2: Find two numbers that multiply to ac and add to b

Look for two numbers that multiply to -70 and add to 9. The numbers are 14 and -5 because \[14 \times (-5) = -70\] and \[14 + (-5) = 9\].

Step 3: Rewrite the middle term

Replace the middle term (9x) with the two numbers found: \[2x^2 + 14x - 5x - 35\].

Step 4: Factor by grouping

Group terms into pairs: \[(2x^2 + 14x) + (-5x - 35)\].Factor out the common factors in each group: \[2x(x + 7) - 5(x + 7)\].

Step 5: Factor out the common binomial

Factor out the common binomial factor \[(x + 7)\]: \[(2x - 5)(x + 7)\].

Step 6: Check the factoring

Verify by expanding \[(2x - 5)(x + 7)\] to ensure it simplifies back to the original expression: \[2x^2 + 14x - 5x - 35\], which indeed simplifies to \[2x^2 + 9x - 35\].

Key Concepts

quadratic equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0 where a, b, and c are constants, and a ≠ 0. The equation represents a parabola when graphed. Quadratic equations can be solved using various methods including factoring, completing the square, and using the quadratic formula.

Solving quadratic equations often involves finding the values of x (roots) that satisfy the equation. These roots can be real or complex numbers.

In the context of our exercise, we are dealing with a quadratic expression (specifically, 2x^2 + 9x - 35 ), and our goal is to factor it using a specific method, known as the ac method.

factoring polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is particularly useful for simplifying polynomial expressions and solving polynomial equations.

The polynomial 2x^2 + 9x - 35 can be factored into simpler binomial expressions. Factoring requires identifying terms or expressions that, when multiplied together, recreate the original polynomial.

For polynomials involving quadratic terms (ax^2), a common approach to factoring is the ac method, which helps break down complex expressions into more manageable factors.

Practicing factoring enables a better understanding of polynomial structures and provides the skills necessary to solve more complicated mathematical problems.

ac method

The ac method is a structured approach to factoring quadratic polynomials. It involves the following steps:

• Identify the coefficients a, b, and c in the quadratic expression \(ax^2 + bx + c\)
• Multiply a and c. In our example, 2 × -35 = -70
• Find two numbers that multiply to ac and add up to b. Here, the numbers are 14 and -5 because 14 × -5 = -70 and 14 - 5 = 9
• Rewrite the middle term (bx) using the two found numbers. Replace 9x with 14x - 5x to get \(2x^2 + 14x - 5x - 35\)
• Group the terms into pairs and factor out common factors: \((2x^2 + 14x) + (-5x - 35)\) becomes \(2x(x + 7) - 5(x + 7)\)
• Factor out the common binomial to get \((2x - 5)(x + 7)\)

This results in the polynomial being factored into two simpler binomials.

checking factors

After factoring a polynomial, it is essential to verify that the factors are correct. This step ensures that the factorization process was done accurately.

For our exercise with 2x^2 + 9x - 35 , we factored it into \((2x - 5)(x + 7)\)

To check the factors, expand \((2x - 5)(x + 7)\) and simplify:

\((2x - 5)(x + 7) = 2x(x + 7) - 5(x + 7) = 2x^2 + 14x - 5x - 35 = 2x^2 + 9x - 35 \) which is the original quadratic expression. This confirms our factorization is correct.

Checking factors is a critical step in ensuring accuracy in mathematical problems. It confirms that the original polynomial can indeed be represented by the product of its factors.