Question

Factor the trinomial. $$ 5 x^2+5 x-30 $$

Short answer

The factored form of the trinomial \( 5x^2 + 5x - 30 \) is \( 5(x + 3)(x - 2) \).

Step-by-step solution

Identify the Common Factor

The given trinomial is \(5x^2 + 5x - 30\). Notice that each term has a common factor of 5. Factor out this common factor by dividing each term of the trinomial by the common factor to simplify the trinomial.

Factor Out the Common Factor

Divide each term by 5. This gives \(x^2 + x - 6 \). So, the expression becomes \(5(x^2 + x - 6)\).

Factor the Quadratic Trinomial

Now, try to factor the quadratic trinomial \(x^2 + x - 6\). Start by finding two numbers that multiply to -6 (the constant term) and add to 1 (the coefficient of the x term). These numbers are 3 and -2.

Write the Factors

Write the quadratic trinomial as the product of two binomial factors using the numbers found in Step 3. This gives \((x + 3)(x - 2)\). Substitute back into the equation to give \(5(x + 3)(x - 2)\).

Key concepts

Common Factor

When factoring trinomials or any polynomial, identifying a common factor shared by all terms can significantly simplify the problem. The common factor is a term that divides each term of the polynomial without leaving a remainder.

For example, in the trinomial 5x^2 + 5x - 30, each term is divisible by 5. This is what we call the 'greatest common factor'.

To factor out the common factor:

• Divide each term of the trinomial by 5.
• Write the trinomial as the product of the common factor and the simplified polynomial.

As a result, we get 5(x^2 + x - 6). This initial step is crucial as it makes the subsequent factoring of the quadratic trinomial far easier.

Quadratic Trinomial

Once we've factored out the common factor from our expression, we're left with a quadratic trinomial. A quadratic trinomial is a second-degree polynomial typically written in the form ax^2 + bx + c. After removing the common factor in our example, we have the quadratic trinomial x^2 + x - 6.

To factor it, we look for two numbers that,

• multiply to give the product (ac) of the leading coefficient (a) and the constant term (c).
• add up to the middle coefficient (b).

In our case, we need two numbers that multiply to -6 and add up to 1. These numbers are 3 and -2. Factoring by grouping or using other factoring techniques can then transform the quadratic trinomial into a product of binomials.

Binomial Factors

Binomial factors are two-term expressions that, when multiplied together, reconstruct the original quadratic trinomial. In the final step of our factoring process, we have identified numbers 3 and -2. We use these to write the quadratic trinomial as a product of two binomials:

(x + 3)(x - 2)

Each binomial represents a set of parentheses containing two terms, the x term and the constant, which when multiplied out using the distributive property (FOIL: First, Outer, Inner, Last), give us the original quadratic trinomial. By introducing these binomial factors into our original expression, we have fully factored the trinomial, ending up with 5(x + 3)(x - 2).

Binomial factors are the building blocks of polynomial expressions, and being able to manipulate and factor them is essential for solving and simplifying algebraic expressions.