Problem 41 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 20 z^{2}+7 z+1 $$

Step-by-Step Solution

Answer

The factored form is \((5z + 1)(4z + 1)\).

Step 1: Identify a, b, and c

Identify the coefficients in the trinomial equation: \[ 20z^2 + 7z + 1 \] The coefficient \( a \) for \( z^2 \) is 20, \( b \) for \( z \) is 7, and \( c \) the constant term is 1.

Step 2: Multiply a and c

Multiply the coefficients \( a \) and \( c \), which gives \( 20 \times 1 = 20 \). This product will be used to find two numbers that multiply to 20 and add to \( b = 7 \).

Step 3: Find two numbers

Find two numbers whose product is 20 and whose sum is 7. These numbers are 5 and 4, since \( 5 \times 4 = 20 \) and \( 5 + 4 = 7 \).

Step 4: Rewrite the middle term using the two numbers

Rewrite the trinomial using the two numbers 5 and 4 to break down the middle term:\[ 20z^2 + 5z + 4z + 1 \].

Step 5: Group the terms

Group the terms into two pairs:\[ (20z^2 + 5z) + (4z + 1) \].

Step 6: Factor each group

Factor out the greatest common factor from each group:- From \( 20z^2 + 5z \), factor out 5z: \[ 5z(4z + 1) \].- From \( 4z + 1 \), factor out 1: \[ 1(4z + 1) \].

Step 7: Combine the factors

Both group terms now have a common factor \( (4z + 1) \). Combine the expression as:\[ (5z + 1)(4z + 1) \].

Step 8: Verify the factors

Multiply \( (5z + 1)(4z + 1) \) to ensure it equals the original trinomial:\[ 5z \times 4z + 5z \times 1 + 1 \times 4z + 1 \times 1 = 20z^2 + 5z + 4z + 1 = 20z^2 + 7z + 1 \].

Key Concepts

Polynomials

In mathematics, polynomials are expressions that consist of variables and coefficients. These variables are raised to non-negative integer powers and combined using addition, subtraction, or multiplication. Polynomials are essential in many areas of mathematics because they provide a simple way to represent complex relationships between numbers. They can have one or more terms, and each term is typically a product of a constant (known as the coefficient) and a variable raised to an exponent.

• For example, in the expression \(20z^2 + 7z + 1\), the terms are \(20z^2\), \(7z\), and \(1\).
• The term \(20z^2\) is a product of the coefficient 20 and the variable \(z^2\).

Understanding polynomials is crucial because they form the basis for more advanced mathematical concepts. They appear in everything from simple calculations to complex physics equations. Notably, in algebra, polynomials, particularly trinomials like the one given in the exercise, are incredibly important for modeling real-world situations and solving problems.

Algebra

Algebra is a branch of mathematics that uses symbols and letters alongside numbers to express relationships and solve problems. With algebra, we can move beyond simple arithmetic and begin to understand how different numbers and variables can relate to each other in complex scenarios.
Algebraic expressions can take many forms, and understanding how to manipulate them is fundamental to learning algebra. This exercise involves factoring, a key skill in algebra. Factoring is the process of breaking down an expression into a product of simpler components, which can often simplify solving equations. By representing a polynomial as a product of its factors, we can solve for the variable values that satisfy the equation.

• For instance, in the polynomial \(20z^2 + 7z + 1\), factoring involves rewriting it as \((5z + 1)(4z + 1)\).

Factoring allows us to tackle quadratic equations and find the roots of the polynomial, where the expression equals zero. Mastery of algebraic principles like factoring makes solving complex equations more manageable and less intimidating.

Quadratic Equations

Quadratic equations arise frequently in algebra, represented by expressions of the form \(ax^2 + bx + c = 0\). These equations depict parabolic curves when graphed and have applications in physics, engineering, and more.
In the context of this exercise, \(20z^2 + 7z + 1\) is a trinomial that has been factored to solve for when it equals zero. Solving quadratic equations can be done using various methods, including factoring, completing the square, and using the quadratic formula. Factoring is often the simplest method if it is possible, because it directly reveals the solutions or 'roots' of the equation.

• When you factor the equation \(20z^2 + 7z + 1\) into \((5z + 1)(4z + 1)\), you can solve for \(z\) by setting each factor equal to zero: \(5z + 1 = 0\) or \(4z + 1 = 0\).
• This yields solutions for \(z\), which are the points where the graph of the equation crosses the \(z\)-axis.

The ability to factor quadratic equations and find their solutions is a key step in advanced algebra, leading students toward understanding more intricate functions and mathematical modeling.