Factoring Quadratic Expressions: \(x^2 - x - 132\)

Understanding how to factor quadratic expressions is a fundamental skill in algebra. The goal is to rewrite the expression as a product of two or more simpler expressions, usually binomials.

The Quadratic Expression to Factor

We need to find the factors of the expression: \(x^2 - x - 132\).

This is a quadratic trinomial in the standard form \(ax^2 + bx + c\), where \(a = 1\), \(b = -1\), and \(c = -132\).

Method for Factoring \(x^2 + bx + c\) where \(a=1\)

To factor a quadratic expression of the form \(x^2 + bx + c\), we look for two numbers, let's call them \(p\) and \(q\), such that:

• Their product is equal to the constant term \(c\): \(p \times q = c\).
• Their sum is equal to the coefficient of the middle term \(b\): \(p + q = b\).

If we find such numbers \(p\) and \(q\), the factored form of the expression is \((x + p)(x + q)\).

Finding the Numbers \(p\) and \(q\) for \(x^2 - x - 132\)

For our expression \(x^2 - x - 132\):

• We need two numbers \(p\) and \(q\) that multiply to \(c = -132\).
• We need these same two numbers to add up to \(b = -1\).

Since the product \(p \times q\) is negative (-132), one of the numbers must be positive, and the other must be negative. Since the sum \(p + q\) is negative (-1), the number with the larger absolute value must be negative.

Let's list pairs of factors of 132 and check their difference (since one factor will be positive and the other negative, their sum will be their difference with the sign of the larger number):

The pair of numbers that multiply to -132 and add up to -1 are -12 and 11. So, \(p = 11\) and \(q = -12\) (or vice versa).

Writing the Factored Form

Using the numbers \(p=11\) and \(q=-12\), the factored form is \((x + p)(x + q) = (x + 11)(x + (-12))\), which simplifies to \((x + 11)(x - 12)\).

Alternative Method: Factoring by Grouping

We can also factor by splitting the middle term \( -x \) using the numbers we found, -12 and 11:

\(x^2 - x - 132\)

Rewrite the middle term: \(x^2 - 12x + 11x - 132\)

Group the terms: \((x^2 - 12x) + (11x - 132)\)

Factor out the greatest common factor (GCF) from each group:

• From \((x^2 - 12x)\), the GCF is \(x\). Factoring it out gives \(x(x - 12)\).
• From \((11x - 132)\), the GCF is 11 (since \(132 = 11 \times 12\)). Factoring it out gives \(11(x - 12)\).

So the expression becomes: \(x(x - 12) + 11(x - 12)\)

Now, notice that \((x - 12)\) is a common binomial factor. Factor out \((x - 12)\):

\((x - 12)(x + 11)\)

This confirms our previous result.

Comparing with Options

Let's look at the options provided and compare them to our factored form \((x - 12)(x + 11)\):

1. \((x - 11)(x - 12)\)
2. \((x + 12)(x - 11)\)
3. \((x + 11)(x + 12)\)
4. \((x - 12)(x + 11)\)

Our factored form \((x - 12)(x + 11)\) matches option 4. Note that the order of factors does not matter, so \((x + 11)(x - 12)\) is the same as \((x - 12)(x + 11)\).

Revision Table: Factoring \(x^2 - x - 132\)

Additional Information on Quadratic Factoring

Factoring quadratics is useful for solving quadratic equations, simplifying expressions, and working with parabolas.

• Difference of Squares: A special case is factoring expressions like \(a^2 - b^2\), which factors into \((a - b)(a + b)\).
• Perfect Square Trinomials: Expressions like \(a^2 + 2ab + b^2\) factor into \((a + b)^2\), and \(a^2 - 2ab + b^2\) factor into \((a - b)^2\).
• Factoring when \(a \neq 1\): For \(ax^2 + bx + c\) where \(a \neq 1\), you look for numbers that multiply to \(ac\) and add to \(b\), then factor by grouping.
• Prime Polynomials: Not all quadratic expressions can be factored into binomials with integer coefficients. These are called prime polynomials. You can check the discriminant (\(b^2 - 4ac\)) to see if real roots exist, which relates to factorability.

Practicing various types of factoring will help you become proficient.