Problem 32 - Elementary Algebra

Question

Factor. $$ 3 x 2+39 x-90 $$

Step-by-Step Solution

Answer

The factored form is \(3(x + 15)(x - 2)\).

Step 1: Identify Common Factors

First, check if there is a greatest common factor (GCF) for all terms in the expression. The terms are \(3x^2\), \(39x\), and \(-90\). The GCF of these coefficients (3, 39, -90) is 3. We factor 3 out of the expression.

Step 2: Factor Out the Common Factor

After identifying 3 as the GCF, we factor it out from the expression: \(3(x^2 + 13x - 30)\). This is our expression after taking out the GCF.

Step 3: Factor the Quadratic Expression

Now we must factor the quadratic expression \(x^2 + 13x - 30\). We need two numbers that multiply to \(-30\) (the constant term) and add to \(13\) (the linear coefficient). The numbers \(15\) and \(-2\) satisfy this condition since \(15 \times -2 = -30\) and \(15 + (-2) = 13\).

Step 4: Express as a Product of Binomials

Use the numbers found in the previous step to write the quadratic expression as a product of two binomials: \((x + 15)(x - 2)\). Thus, the expression becomes: \(3(x + 15)(x - 2)\).

Step 5: Verify the Factorization

To ensure the factorization is correct, expand \(3(x + 15)(x - 2)\) and check if it equals the original expression. Expanding gives:\[3(x + 15)(x - 2) = 3[(x)(x) + (x)(-2) + (15)(x) + (15)(-2)] = 3(x^2 - 2x + 15x - 30) = 3(x^2 + 13x - 30) = 3x^2 + 39x - 90\].Since this equals the original expression, the factorization is verified.

Key Concepts

Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is an essential first step in factoring any expression. Consider the expression consisting of multiple terms, like coefficients or variables, and search for the largest number that divides each of these terms without a remainder.For the quadratic expression given: \(3x^2 + 39x - 90\), the coefficients 3, 39, and -90 are obvious candidates.Determine their greatest common factor by identifying the largest number that divides into each term.In this case, it's 3. Once identified, it's essential to factor out this GCF from the entire expression.By factoring out 3, it simplifies to \(3(x^2 + 13x - 30)\).

• Examining each term for commonality simplifies the expression.
• Factoring out the GCF always simplifies calculations and reveals the underlying structure of the polynomial.

Always determine the GCF first to make further steps easier and more straightforward.

Quadratic Polynomials

Quadratic polynomials are expressions where the highest power of the variable is two, appearing as \(ax^2 + bx + c\).The quadratic expression left after factoring out the GCF is \(x^2 + 13x - 30\).Here, our task is to factor it further into simpler expressions.Quadratic polynomials often present a challenge in factoring because they require finding two numbers that satisfy two conditions:

• The product of those numbers equals the constant term multiplied by the leading coefficient (\(a \times c\)).
• The sum of those numbers must equal the linear coefficient (\(b\)).

For \(x^2 + 13x - 30\), the mission is simple:Find two numbers that multiply to -30 and sum to 13.The numbers 15 and -2 work perfectly because \(15 \times -2 = -30\) and \(15 + (-2) = 13\).This breakthrough allows you to express the quadratic polynomial as a product of binomials.

Binomial Products

Binomial products result from expressing a quadratic equation as two simpler linear expressions, more commonly known as binomials. In factoring the quadratic \(x^2 + 13x - 30\), the numbers 15 and -2 guide the creation of the binomials: \((x + 15)(x - 2)\).These new expressions show that the original quadratic breaks down into two factors, each with two terms.This binomial representation allows for easier manipulation and further understanding of the polynomial's roots.

• Each binomial represents a possible solution where the overall expression equals zero.
• Factoring into binomials is essential for solving quadratic equations and finding the values of \(x\) where the equation equals zero.

Once the binomial products are identified, the expression \(3(x + 15)(x - 2)\) showcases a fully factored form of the original quadratic.By expanding the binomials, you can verify the results reliably.The transition from expression to verified factorization provides confidence in the algebraic manipulation undertaken.