Problem 31
Question
Factor completely each of the polynomials and indicate any that are not factorable using integers. $$8 x^{2}+26 x-45$$
Step-by-Step Solution
Verified Answer
The polynomial factors completely as \((4x - 5)(2x + 9)\).
1Step 1: Identify the A, B, and C values
The polynomial is in the form \( ax^2 + bx + c \). Identify the coefficients: \( a = 8 \), \( b = 26 \), and \( c = -45 \).
2Step 2: Determine the product of A and C
Multiply \( a \) and \( c \) to find the product: \( 8 imes (-45) = -360 \).
3Step 3: Find two numbers that multiply to AC and add to B
We need two numbers that multiply to \(-360\) and add to \(26\). These numbers are \(36\) and \(-10\) because \(36 imes (-10) = -360\) and \(36 + (-10) = 26\).
4Step 4: Rewrite the middle term
Rewrite \(26x\) as \(36x - 10x\). The polynomial now is \( 8x^2 + 36x - 10x - 45 \).
5Step 5: Group terms
Group the terms: \( (8x^2 + 36x) + (-10x - 45) \).
6Step 6: Factor out the greatest common factor (GCF) from each group
Factor \(4x\) from the first group and \(-5\) from the second group:\( 4x(2x + 9) - 5(2x + 9) \).
7Step 7: Factor out the common binomial factor
Both groups contain the common factor \((2x + 9)\). Factor this out:\( (4x - 5)(2x + 9) \).
8Step 8: Final check of factors
Verify the factors: Expand \((4x - 5)(2x + 9)\) to check if the original polynomial \(8x^2 + 26x - 45\) is obtained.
Key Concepts
Quadratic EquationsGreatest Common Factor (GCF)Binomial Factors
Quadratic Equations
Quadratic equations are very common in algebra and appear widely in mathematics. They take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. The solutions to a quadratic equation are the values of \( x \) that make the equation true. These solutions can be found by factoring, using the quadratic formula, or by completing the square.
Many problems in physics, engineering, and other sciences can be modeled by quadratic equations. Solving these equations often yields values that indicate maximum or minimum points, projectile paths, or optimal points in a strategy.
The process of factoring quadratics involves expressing the equation as a product of its factors, such as \( ax^2 + bx + c = (dx + e)(fx + g)\), where \( d \), \( e \), \( f \), and \( g \) are determined such that the product correctly expands back to the original quadratic equation. Students typically practice this through various algebraic techniques, and it requires a keen eye for working with numbers and patterns.
Many problems in physics, engineering, and other sciences can be modeled by quadratic equations. Solving these equations often yields values that indicate maximum or minimum points, projectile paths, or optimal points in a strategy.
The process of factoring quadratics involves expressing the equation as a product of its factors, such as \( ax^2 + bx + c = (dx + e)(fx + g)\), where \( d \), \( e \), \( f \), and \( g \) are determined such that the product correctly expands back to the original quadratic equation. Students typically practice this through various algebraic techniques, and it requires a keen eye for working with numbers and patterns.
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest positive integer that can divide two or more numbers without leaving a remainder. It plays a critical role when simplifying expressions and equations by allowing us to factor out the most common factor from a group of terms.
In the context of polynomials, finding the GCF eases the factoring process. For example, when you have terms like \(8x^2\) and \(36x\), finding the GCF, in this case, \(4x\), means you can simplify by pulling \(4x\) out of the expression, leading to a less complex polynomial form to work with.
In the context of polynomials, finding the GCF eases the factoring process. For example, when you have terms like \(8x^2\) and \(36x\), finding the GCF, in this case, \(4x\), means you can simplify by pulling \(4x\) out of the expression, leading to a less complex polynomial form to work with.
- Using GCF reduces equations down to a simpler equivalent, making subsequent steps more manageable.
- It can be viewed as a method for de-cluttering expressions before diving into more complex operations, such as solving for variables or further factoring.
Binomial Factors
Binomial factors are expressions that consist of two terms separated by a plus or minus sign, such as \((x+3)\) or \((2x-9)\). These are the building blocks in the process of factoring polynomials and play a decisive role in solving quadratic equations.
In the context of factoring polynomials, as seen in the solution for the given exercise, the polynomial was rewritten to reveal binomial factors. The expression \(4x(2x + 9) - 5(2x + 9)\) was transformed by recognizing the common binomial \((2x+9)\). This factor is often called the 'common factor' because it appears in both terms being grouped.
In the context of factoring polynomials, as seen in the solution for the given exercise, the polynomial was rewritten to reveal binomial factors. The expression \(4x(2x + 9) - 5(2x + 9)\) was transformed by recognizing the common binomial \((2x+9)\). This factor is often called the 'common factor' because it appears in both terms being grouped.
- Using binomial factor recognition helps streamline the simplification process.
- It allows for the original polynomial to be expressed in a fully factored form, which can expose roots or simplify equations under certain conditions.
Other exercises in this chapter
Problem 30
Subtract the polynomials using the horizontal format. \(6 x^{3}+x^{2}+4\) from \(9 x^{3}-x-2\)
View solution Problem 31
Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$15 x^{2}+34 x+15=0$$
View solution Problem 31
Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial facto
View solution Problem 31
Factor completely. $$18 a^{2} b+27 a b^{2}$$
View solution