Problem 25 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$14 y^{2}+15 y-9$$

Step-by-Step Solution

Answer

The factorised form of the trinomial \(14y^{2}+15y-9\) is \(7y(2y + 3) - 3(2y + 3)\).

Step 1: Identify the structure of the trinomial

The given trinomial is a quadratic trinomial since the highest degree of the variable (y in this case) is 2. It's in the form \(ay^{2} + by + c\), where a=14, b=15, and c=-9.

Step 2: Factorize the trinomial

To factorize the trinomial, search for two numbers such that when multiplied the result is a*c (product of a and c) and when added together the result is b. The product of a and c is 14*(-9) = -126. The numbers that give -126 when multiplied and 15 when added together are 21 and -6. Therefore, the trinomial \(14y^{2}+15y-9\) can be rewritten as: \(14y^{2} + 21y - 6y - 9\). This equation can be broken into two groups: \(14y^{2} + 21y\) and \(-6y - 9\). We can factorise further by taking out the common factor from each binomial. For the first binomial group, we can factorise by taking out the common factor 7y. This leaves us with \(7y(2y + 3)\). For the second binomial group, we can factorise by taking out the common factor -3. This leaves us with \(-3(2y + 3)\). The final factorised form is then: \(7y(2y + 3) - 3(2y + 3)\).

Step 3: Check the factorization using FOIL.

To verify the correctness of this factorization, expand the brackets using the FOIL - First, Outer, Inner, Last - principle. Thus, \(7y * 2y = 14y^{2}\). \(7y * 3 = 21y\). \(-3 * 2y = -6y\). \(-3 * 3 = -9\). Adding this all gives back the original trinomial. \(14y^{2} + 21y - 6y - 9 = 14y^{2} + 15y - 9\). Therefore, the check has confirmed the correctness of the factorization.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are expressions in mathematics that take the form of a second-degree polynomial, typically written as \( ax^2 + bx + c \). Key features include the highest exponent being 2, indicating the presence of a quadratic term, a linear term, and a constant.

Factoring such trinomials involves finding two binomial expressions that, when multiplied together, return the original trinomial. This process is crucial because it transforms the quadratic trinomial into a product of simpler expressions, making it easier to solve equations or find the x-intercepts of a function.

Understanding the Coefficients

When working with quadratic trinomials, the coefficients represent integral parts: 'a' is the coefficient of the quadratic term \(x^2\), 'b' is the coefficient of the linear term (x), and 'c' is the constant. The values of these coefficients affect the factoring strategy and must be carefully considered in the process.

FOIL Method

The FOIL method is a technique used to multiply two binomials and stands for First, Outer, Inner, Last. This mnemonic helps students remember the order in which to multiply the terms.

To apply FOIL, multiply the First terms of each binomial together, then the Outer terms, followed by the Inner terms, and finally, the Last terms. Once these products are obtained, they are summed to get the expanded polynomial.

Application of FOIL

Using the FOIL method is essential when checking the accuracy of the factored form of a quadratic trinomial. After factoring, one should always multiply the binomials to ensure that the original expression is obtained, confirming that the factoring process has been done correctly.

Binomial Factorization

Binomial factorization involves breaking down a quadratic trinomial into the product of two binomials. This process often requires finding a common factor or applying special patterns like difference of squares or perfect square trinomials.

When factorizing, you look for two binomials that are multiplied in such a way that when expanded they return the original trinomial.

Strategies for Factoring

If the trinomial has a leading coefficient of one (\(a=1\)), you search for two numbers that multiply to the constant term and add to the linear coefficient. For a trinomial with a leading coefficient greater than one, like in the given exercise, it's more complex and involves finding a pair of numbers that multiply to \(ac\) and add to \(b\). After splitting the middle term based on these two numbers, grouping and factoring by grouping are applied to arrive at the binomial factors.