Problem 101 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. (Lessons 10.5,10.6) $$ x^{2}+x-20 $$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}+x-20\) is \((x-4)(x+5)\).

Step 1: Identify \(a\), \(b\) and \(c\) in the trinomial

Compare the given trinomial with the quadratic expression \(ax^2 + bx + c\). Here \(a=1, b=1, c=-20\) correspondingly.

Step 2: Find two numbers that multiply to \(ac\) and add to \(b\)

Calculate two numbers that multiply to \(ac=-20\) and add to \(b=1\). Here, the numbers are +5 and -4 because \(5*-4=-20\) and \(5+-4=1\).

Step 3: Rewrite the trinomial using these numbers

The trinomial \(x^2+x-20\) can be rewritten as \(x^2+5x-4x-20\).

Step 4: Factor by grouping

Group the terms in the trinomial and then factor by grouping. This gives us \((x^2+5x)-4(x+5)\). Here we take out \(x\) common from \(x^2+5x\) and -4 common from -4x-20.

Step 5: Complete the factorization

Completing the factorization gives \(x(x+5)-4(x+5)\).

Step 6: Write the final factored form

We now write the final factored form, which is \((x-4)(x+5)\).

Key Concepts

Quadratic Expressions

A quadratic expression is a polynomial of degree 2, generally in the form of \(ax^2 + bx + c\). These are used often in algebra and represent the equation of a parabola when plotted on a graph. In mathematics, these expressions enable us to find solutions that give roots, intercepts, or explain patterns depending on the context.
Understanding quadratic expressions is vital because they form the basis of factoring and solving equations. Unlike linear expressions, quadratic expressions can have two solutions or roots. This occurs because they involve a squared term, introducing more complexity.
To work with quadratic expressions, get comfortable with terms like "standard form", "roots" (the solutions), "factors" (numbers that multiply to give the original expression), and remembering that the highest degree of the variable here is 2.

Factor by Grouping

Factor by grouping is an especially useful method to factor quadratic polynomials when the leading coefficient \(a\) is not 1, or when the trinomial cannot be easily factored. This method involves rearranging and grouping terms in a way that makes factoring simpler.
In our problem \(x^2 + x - 20\), we initially rewrite it as \(x^2 + 5x - 4x - 20\). This step might seem odd at first, but it helps in setting up the expression for grouping. We grouped them as \((x^2 + 5x)\) and \((-4x - 20)\), then look for common factors.
By extracting these common factors from each group,\(x\) becomes the common factor of the first group and \(-4\) the common factor of the second group.
This method may take practice, as finding the right numbers to split "b" and the right factors requires some trial and error. However, it's incredibly powerful once mastered.

Rewriting Expressions

Rewriting expressions is a strategic method used to simplify, solve, or understand equations better. By altering the expression's form, we can often make complex problems more manageable.
In our exercise, we took \(x^2 + x - 20\) and rewrote it as \(x^2 + 5x - 4x - 20\). The reason behind rewriting is to break the original expression into parts that are easy to group and factor.
During this, your goal is to find a pair of numbers that multiply to the product of \(a\times c\) and add up to \(b\). Mastering this technique strengthens algebraic understanding and problem-solving skills by allowing manipulation of expressions to reveal deeper insights or hidden relationships, which are crucial in mathematics.
Incorporating rewriting in algebra is like rearranging pieces of a puzzle until they fit just right, revealing a clearer picture of the solution.