Problem 13 - Algebra and Trigonometry

Question

\(13-20\) . Factor the trinomial. $$ x^{2}+2 x-3 $$

Step-by-Step Solution

Answer

The trinomial \(x^2 + 2x - 3\) factors to \((x + 3)(x - 1)\).

Step 1: Identify Coefficients

The given trinomial is \(x^2 + 2x - 3\). To factor it, first identify the coefficients: \(a = 1\), \(b = 2\), and \(c = -3\).

Step 2: Find Product of Coefficient 'a' and 'c'

Calculate the product of the coefficients \(a\) and \(c\), which is \(1 \times -3 = -3\).

Step 3: Determine Factors of Product

Find two numbers that multiply to \(-3\) (the result from Step 2) and add to \(2\) (the coefficient \(b\)). The numbers \(3\) and \(-1\) satisfy these conditions because \(3 \times -1 = -3\) and \(3 + (-1) = 2\).

Step 4: Rewrite the Middle Term

Rewrite the trinomial by splitting the middle term using the numbers found in Step 3: \[x^2 + 3x - x - 3\].

Step 5: Group Terms

Group the terms into two pairs: \[(x^2 + 3x) + (-x - 3)\].

Step 6: Factor Each Group

Factor out the greatest common factor from each group: - For \((x^2 + 3x)\), factor out \(x\): \(x(x + 3)\).- For \((-x - 3)\), factor out \(-1\): \(-1(x + 3)\).

Step 7: Factor Out the Common Binomial

Notice that \((x+3)\) is a common factor. Factor it out: \[(x + 3)(x - 1)\].

Step 8: Verify the Solution

Expand the factors to verify they simplify back to the original trinomial: \[(x+3)(x-1) = x^2 - x + 3x - 3 = x^2 + 2x - 3\]. The expansion is the original trinomial, confirming the factorization is correct.

Key Concepts

Trinomial Coefficients

Understanding the concept of trinomial coefficients is crucial when it comes to factoring trinomials. Trinomials are algebraic expressions consisting of three terms. Here, we're dealing with a quadratic trinomial which is of the form \(ax^2 + bx + c\). In this format:

• \(a\) is the coefficient of the quadratic term \(x^2\).
• \(b\) is the coefficient of the linear term \(x\).
• \(c\) is the constant term.

In our exercise, the trinomial \(x^2 + 2x - 3\) can be broken down to:

• \(a = 1\)
• \(b = 2\)
• \(c = -3\)

Identifying these coefficients is the first step in the factoring process, helping us determine which numbers we need to use to split the middle term effectively.

Factoring by Grouping

Factoring by grouping is a technique often employed to factor quadratics, especially when simple factoring isn't straightforward. It involves grouping terms in a way that allows for common factors to be factored out easily. In our trinomial, after finding suitable numbers that multiply to the product of \(a\) and \(c\) and add to \(b\), we rewrite the expression to make grouping possible.For \(x^2 + 2x - 3\), we previously determined that 3 and -1 meet our conditions. So, we rewrite the trinomial as \(x^2 + 3x - x - 3\). Group these terms into pairs:

• \((x^2 + 3x)\) and
• \((-x - 3)\)

Each group reveals a common factor: - Factor out \(x\) from \(x^2 + 3x\) to get \(x(x+3)\).- Factor out \(-1\) from \(-x - 3\) to get \(-1(x+3)\).Notice now how the common binomial \((x+3)\) simplifies the expression into a product of binomials.

Polynomial Factorization

Polynomial factorization is the process of breaking a polynomial down into simpler "factors" that, when multiplied, give the original polynomial. For quadratic expressions, this often results in two binomials. In our example:- After factoring by grouping, we end up with two smaller expressions: \(x(x+3)\) and \(-1(x+3)\).- Since \((x+3)\) is a common factor, it can be factored out, leaving us with \((x+3)(x-1)\).This process highlights how understanding the factorization of polynomials can simplify and solve quadratic expressions, making them easier to manage. It not only helps in algebra but is also fundamental in solving quadratic equations and understanding complex functions.

Quadratic Expressions

Quadratic expressions form the basis of many algebraic equations and appear frequently across mathematics. These expressions are characterized by the variable being squared, hence the name "quadratic" which doubles up as the Latin word for square.In the expression \(x^2 + 2x - 3\), quadratic expressions give us a curve called a parabola when graphed. Factoring them simplifies solutions to equations, and reveals the roots or solutions to the equation \(x^2 + 2x - 3 = 0\).The roots we find, in this case, \((x+3)\) and \((x-1)\), correspond to:

• Solving \(x+3=0\) gives \(x=-3\)
• Solving \(x-1=0\) gives \(x=1\)

Understanding quadratic expressions paves the way for more advanced topics in algebra, such as completing the square, and using the quadratic formula. It forms a bridge to more complex polynomial equations.