Question

Factor each polynomial. $$ 3 x^{2}-10 x+8 $$

Short answer

\((3x - 4)(x - 2)\)

Step-by-step solution

Identify Coefficients

Identify the coefficients of the polynomial. Here, the polynomial is given by: \(3x^2 - 10x + 8\)The coefficient for \(x^2\) is 3, for \(x\) it is -10, and the constant term is 8.

Find Product and Sum

Find two numbers that multiply to the product of the coefficient of \(x^2\) and the constant term (3 * 8 = 24) and add up to the coefficient of x (-10). These numbers are -4 and -6 because (-4) * (-6) = 24 and (-4) + (-6) = -10.

Rewrite the Middle Term

Rewrite the middle term \(-10x\) using the two numbers found: \(3x^2 - 4x - 6x + 8\)

Factor by Grouping

Group the terms in pairs and factor out the common factor from each pair: \(3x^2 - 4x - 6x + 8 = x(3x - 4) - 2(3x - 4)\)

Factor the Common Binomial Factor

Factor out the common binomial factor \((3x - 4)(x - 2)\)

Key concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2. They have the general form: \[ ax^2 + bx + c = 0 \] where:

• \(a\), \(b\), and \(c\) are constants
• \(x\) is the variable

In the given problem, the quadratic equation is \(3x^2 - 10x + 8\). We aim to factorize this into two binomials. Understanding and solving quadratic equations is essential because they frequently appear in algebra, physics, engineering, and many other fields.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. For quadratic polynomials, we typically look for two binomials. Here is the step-by-step process:

1. Identify coefficients: Locate the coefficients in the polynomial. For \(3x^2 - 10x + 8\), they are 3, -10, and 8.
2. Product and sum method: Find two numbers that multiply to \(3 * 8 = 24\) and add to the middle coefficient, -10. The numbers are -4 and -6.
3. Rewrite the polynomial: Rewrite the middle term, \(-10x\), using these numbers to get \(3x^2 - 4x - 6x + 8\).
4. Grouping: Group the terms to facilitate factoring: \( (3x^2 - 4x) + (-6x + 8) \).
5. Factor out common factors: Factor out the GCD (greatest common divisor) from each group: \( x(3x - 4) - 2(3x - 4) \).
6. Factor common binomials: Combine the common binomials to get the final factored form: \( (3x - 4)(x - 2) \).

By following these steps, we ensure that the polynomial is correctly factored.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operations (addition, subtraction, multiplication, and division). For instance, in the polynomial \(3x^2 - 10x + 8\), different parts are broken down as follows:

• \(3x^2\): This term indicates that 3 is multiplied by \(x\) squared.
• \(-10x\): This means 10 is multiplied by \(x\) with a negative sign.
• 8: A constant term.

When working with algebraic expressions, your goal is to simplify or manipulate them to a form that's easier to work with. In the above problem, we simplified the expression by factoring.

Coefficient Identification

Coefficients are the numerical parts of terms with variables in an algebraic expression. In our polynomial \(3x^2 - 10x + 8\), the coefficients are:

• 3 for \(x^2\)
• -10 for \(x\)
• 8 as the constant term

Identifying coefficients is the first step in polynomial factoring. They help us understand the relationships between terms and perform operations like the product and sum method used in factoring quadratics. By knowing the coefficients, you can easily break down and manipulate polynomials to solve algebraic equations efficiently.