Problem 23 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 4 x^{2}-8 x-21 $$

Step-by-Step Solution

Answer

The factored form is \((4x - 21)(x + 1)\).

Step 1: Identify the Quadratic Polynomial

The expression is a quadratic trinomial: \(4x^2 - 8x - 21\). We need to factor it using the grouping method.

Step 2: Multiply the Leading Coefficient and the Constant Term

Identify the leading coefficient (4) and the constant term (-21). Multiply them together: \(4 \times -21 = -84\).

Step 3: Find Two Numbers that Multiply to -84 and Add to Middle Term

Look for two numbers whose product is -84 and whose sum is the middle coefficient, -8. The numbers are 4 and -21, since \(4 \times -21 = -84\) and \(4 + (-21) = -17\).

Step 4: Split the Middle Term using the Identified Numbers

Rewrite the middle term, -8x, using the numbers found in step 3: \(4x^2 + 4x -21x - 21\).

Step 5: Group the Terms

Separate the expression into two groups: \((4x^2 + 4x) - (21x + 21)\).

Step 6: Factor Out GCD in Each Group

Factor the greatest common divisor from each group. Factor 4x from the first group and -21 from the second: \(4x(x + 1) - 21(x + 1)\).

Step 7: Factor the Common Binomial

Notice that \(x + 1\) is common in both groups. Factor it out: \((4x - 21)(x + 1)\).

Step 8: Final Step: Simplified Factored Form

The trinomial \(4x^2 - 8x - 21\) is factored as \((4x - 21)(x + 1)\).

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a mathematical expression of the form \( ax^2 + bx + c \), where "a," "b," and "c" are constants. Quadratic polynomials are called 'quadratic' because the highest degree term, \( x^2 \), is a square or 'quad.' The equation given, \( 4x^2 - 8x - 21 \), is a classic example of a quadratic polynomial.
Understanding the structure of quadratic polynomials helps in many algebraic applications. These expressions often appear in problems involving motion, area, and optimization.
The process of factoring a quadratic polynomial involves breaking it down into simpler expressions that can be multiplied to give the original polynomial. Factoring is a critical skill as it simplifies equations and makes solving them much easier.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. In a quadratic polynomial like \( ax^2 + bx + c \), the leading coefficient is "a." In this exercise, the leading coefficient is 4, coming from the term \( 4x^2 \).
Understanding the leading coefficient is important when factoring polynomials. It affects the process because it controls the behavior of the polynomial's graph and changes how we approach factoring.
When factoring by grouping, one starts by multiplying the leading coefficient with the constant term. This product, in this case \( 4 \times -21 = -84 \), plays a crucial role in deciding how to split the middle term.

Greatest Common Divisor

The greatest common divisor (GCD) of a set of terms is the largest number that divides each of the terms without leaving a remainder. It is an essential tool in factoring because it allows extraction of common factors from terms.
In the process of factoring the trinomial \( 4x^2 - 8x - 21 \) by grouping:

• We grouped the terms into \((4x^2 + 4x)\) and \((-21x - 21)\).
• The GCD for the first group \((4x^2 + 4x)\) is 4x, since both terms can be divided by 4x.
• For the second group \((-21x - 21)\), the GCD is -21, allowing each term to be divided by -21.

Factoring out the GCD helps simplify the expression into a product involving a common binomial factor. This is the key to achieving the factored form efficiently.

Binomial Factor

A binomial factor is an expression with two terms that can be multiplied, added, or subtracted within an algebraic structure. In factoring, identifying common binomial factors can significantly simplify complex expressions.
In our exercise, after factoring out the GCD from the two grouped parts, we ended with expressions \( 4x(x + 1) \) and \(-21(x + 1) \). Here, \( x + 1 \) is the repeated binomial factor.
Recognizing that \( x + 1 \) appears in both groups allows us to factor it out from the entire expression. This gives the final simplified form \( (4x - 21)(x + 1) \), effectively completing the factoring process.
Mastery of identifying binomial factors through grouping and extraction is vital in solving quadratic equations efficiently.