Problem 13 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$8 x^{2}-53 x-21$$

Step-by-Step Solution

Answer

The factors of the polynomial are \((8x + 3)(x - 7)\).

Step 1: Identify the polynomial

The given polynomial is \(8x^2 - 53x - 21\). This is a quadratic polynomial of the form \(ax^2 + bx + c\), where \(a = 8\), \(b = -53\), and \(c = -21\).

Step 2: Multiply 'a' and 'c'

Multiply the coefficients \(a\) and \(c\). Here, you calculate \(8 \times (-21) = -168\).

Step 3: Find two numbers that multiply and add (or subtract)

Look for two numbers that multiply to \(-168\) and add to \(-53\). These numbers are \(-56\) and \(3\).

Step 4: Rewrite the middle term

Rewrite \(-53x\) with the numbers found: \(8x^2 - 56x + 3x - 21\).

Step 5: Factor by grouping

Group the terms: \((8x^2 - 56x) + (3x - 21)\). Factor each group: \(8x(x - 7) + 3(x - 7)\).

Step 6: Factor out the common factor

Notice \((x - 7)\) is common to both groups. Factor it out: \((8x + 3)(x - 7)\).

Step 7: Verify the factors

Expand \((8x + 3)(x - 7)\) to ensure it equals the original polynomial: \[8x^2 - 56x + 3x - 21 = 8x^2 - 53x - 21\]. The factors are correct.

Key Concepts

Quadratic Polynomials

A quadratic polynomial is any polynomial of degree 2. It typically takes the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The degree of the polynomial is determined by the highest power of \(x\), which in the case of a quadratic is 2.

Quadratic polynomials can arise in various contexts such as physics, engineering, and statistics, and they often model situations such as projectile paths and area calculations.

Key characteristics of quadratic polynomials include the presence of a parabolic graph, which can either open upwards (if \(a > 0\)) or downwards (if \(a < 0\)). The solutions or "roots" of a quadratic polynomial are the values of \(x\) for which the polynomial is equal to zero.

Understanding the structure of quadratics helps in solving them, either by factorization, completing the square, or using the quadratic formula.

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of its factors. For quadratic polynomials, this means expressing them in the form \((dx + e)(fx + g)\), where the product gives back the original quadratic expression.

The process of factorization simplifies the polynomial and makes it easier to find its roots. Factorization essentially breaks down the polynomial into smaller, simpler expressions that reveal valuable insights about the polynomial's behavior at specific values of \(x\).

Common methods of factorization include:

• Factoring by grouping
• Using special formulas like the difference of squares, perfect square trinomials
• Utilizing synthetic division for higher-degree polynomials

Factorization is crucial in algebra since it provides a direct method to solve equations and explore polynomial properties.

Factoring by Grouping

Factoring by grouping is a popular technique used particularly when dealing with higher degree polynomials or challenging quadratic expressions. This method involves grouping terms with common factors and simplifying the expression one step at a time.

For example, consider a polynomial like \(8x^2 - 53x - 21\). To factor this by grouping, you would initially need to rearrange and group the terms into pairs that can be factored individually, such as \((8x^2 - 56x)\) and \((3x - 21)\).

Within each group:

• Factor out the greatest common factor (GCF).
• Look for a common factor in the expressions obtained after grouping.

If a common factor is found, it can be factored from the entire expression. This method is especially effective when the polynomial doesn't readily fit known patterns of factorization and needs a strategic approach to break it down.