Problem 17 - 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. $$3 x^{2}-17 x+10$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(3x^{2}-17x+10\) is \((3x-2)(x-5)\).

Step 1: Identifying Coefficients

Recognize the coefficients in the trinomial equation \(3x^{2}-17x+10\). Here, \(a = 3\), \(b = -17\), and \(c = 10\).

Step 2: Factoring the Trinomial

Look for two numbers that both multiplies to \(ac = 3*10 = 30\) and sums up to \(b = -17\). Here, the numbers are -15 and -2, as \(-15*-2 = 30\) and \(-15 - 2 = -17\). Rewrite the equation as: \(3x^{2}-15x-2x+10\). Then, factor by grouping and the factored form of the trinomial will be \((3x-2)(x-5)\).

Step 3: Verifying the Factorization

Verify the solution using the FOIL method. Multiply the first terms, outer terms, inner terms, then last terms separately and add them together to check if they equal the original trinomial equation. Here, \((3x)(x) + (3x)(-5) + (-2)(x) + (-2)(-5) = 3x^{2}-15x-2x+10 = 3x^{2}-17x+10\), which confirms the factored form is correct.

Key Concepts

Trinomial Factorization

Trinomial factorization involves breaking down a trinomial expression into the product of two or more simpler expressions. A trinomial is a type of polynomial with three terms. Consider the case of factoring the trinomial \(3x^{2} - 17x + 10\). The goal is to express this as a product of two binomials.

To begin, identify the coefficients in the trinomial: \(a = 3\), \(b = -17\), and \(c = 10\). Next, we need two numbers that multiply to \(ac = 30\) and add up to \(b = -17\). These numbers are \(-15\) and \(-2\).

With these numbers, rewrite the trinomial as \(3x^{2} - 15x - 2x + 10\), splitting the middle term \(-17x\) into \(-15x\) and \(-2x\). This sets up for the next step: factoring by grouping.

FOIL Method

The FOIL method is a technique used to multiply two binomials. It stands for First, Outer, Inner, and Last, which are the terms from each binomial that you multiply together.

To check the accuracy of factorization, use the FOIL method on the factored form \((3x - 2)(x - 5)\).

• First: Multiply the first terms in each binomial: \(3x \times x = 3x^{2}\).
• Outer: Multiply the outer terms: \(3x \times -5 = -15x\).
• Inner: Multiply the inner terms: \(-2 \times x = -2x\).
• Last: Multiply the last terms: \(-2 \times -5 = 10\).

Adding these results together: \(3x^{2} - 15x - 2x + 10\), you get the original trinomial \(3x^{2} - 17x + 10\).

Using the FOIL method not only helps in multiplication but also serves as a verification step for factorization.

Factoring by Grouping

Factoring by grouping is an efficient strategy for trinomials where the middle term is split into two terms that allow grouping and further factorization.

Taking the trinomial \(3x^{2} - 17x + 10\), use the numbers \(-15\) and \(-2\) to rewrite it as \(3x^{2} - 15x - 2x + 10\). By grouping these into two pairs, \((3x^{2} - 15x)\) and \((-2x + 10)\), you can factor each pair.

• For the first group \((3x^{2} - 15x)\), factor out a \(3x\), resulting in \(3x(x - 5)\).
• For the second group \((-2x + 10)\), factor out a \(-2\), resulting in \(-2(x - 5)\).

Both groups share a common factor \((x - 5)\). Factor out \((x - 5)\), combining them into \((3x - 2)(x - 5)\).

This technique highlights the importance of the strategic splitting of the middle term and grouping terms to simplify complex polynomials.

Quadratic Equations

Quadratic equations are a type of polynomial equation of the second degree, usually in the form \(ax^{2} + bx + c = 0\). The equation \(3x^{2} - 17x + 10\) is an example of a quadratic trinomial.

Solving quadratic equations often involves finding two numbers that satisfy both multiplication and addition criteria, as seen in the factorization process. These numbers help split the middle term for easier grouping and factoring.

Quadratic equations appear frequently in algebra because they describe a wide array of problems. Understanding their structure and methods of solving—like factorization—can open the door to more advanced mathematical topics.
Quadratics can also be solved using formulas such as the quadratic formula, \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), though factorization offers a more intuitive understanding of the relationship between the coefficients and roots of the equation.