Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{2}-10 x+21 $$

Short answer

\((x - 3)(x - 7)\)

Step-by-step solution

Identify the polynomial

Given the polynomial is \(x^{2} - 10x + 21\). It is a quadratic polynomial of the form \(ax^2 + bx + c\) where \(a = 1\), \(b = -10\), and \(c = 21\).

Find two numbers that multiply to \(a \times c\) and add to \(b\)

We need to find two numbers that multiply to \(1 \times 21 = 21\) and add to \(-10\). These numbers are \(-3\) and \(-7\) because \(-3 \times -7 = 21\) and \(-3 + -7 = -10\).

Rewrite the middle term

Rewrite the polynomial by breaking the middle term \(-10x\) using the numbers found: \(x^2 - 3x - 7x + 21\).

Factor by grouping

Group the terms to factor: \((x^2 - 3x) + (-7x + 21)\). Factor out the common factors in each group: \(x(x - 3) - 7(x - 3)\).

Factor out the common binomial

Factor out the common binomial \((x - 3)\), giving the final factored form: \((x - 3)(x - 7)\).

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. The solutions to these equations are known as the roots or zeroes of the equation. Quadratic equations can be solved using different methods, including the quadratic formula, completing the square, and factoring. In this exercise, we are focusing on factoring.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials that are not easily factored by common methods. This technique is useful when dealing with four-term polynomials but can be adapted to other forms.

In the given problem, we rewrite the middle term to break the quadratic polynomial into groups. For example:
\( x^2 - 10x + 21 \) can be split into \( x^2 - 3x - 7x + 21 \).

We then group the terms: \( (x^2 - 3x) + (-7x + 21) \). Each group is factored separately, and their common factors are extracted to help us factor the polynomial completely.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler 'factor' polynomials that when multiplied together, give the original polynomial. For quadratic polynomials, this typically involves expressing the polynomial as a product of two binomials.

In our specific example, we started with the polynomial \( x^2 - 10x + 21 \) and factored it by finding two numbers that multiply to the constant term \(a \times c\) and add up to the coefficient of the linear term \(b\). Through these steps, we factored the polynomial to \((x-3)(x-7)\). This means if you multiply \((x-3)\) and \((x-7)\) together, you will get back the original polynomial.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They can range from simple to complex expressions and often require manipulation to simplify or solve. They are the building blocks of algebra.

In the context of this exercise, we dealt with a quadratic algebraic expression, \(x^2 - 10x + 21\). We aimed to factor this complex expression into simpler binomial factors using various algebraic techniques such as identification of constants, grouping, and factoring out common binomials. Understanding these fundamental techniques is crucial for solving more complex algebraic problems and mastering the subject.