Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 22

Chapter 1, Problem 22

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x²−14x+45

Verified step by step guidance

Identify the trinomial: $x^2 - 14x + 45$. This is a quadratic expression in the form $ax^2 + bx + c$, where $a = 1$, $b = -14$, and $c = 45$.

To factor the trinomial, look for two numbers that multiply to $c = 45$ and add to $b = -14$.

List the factor pairs of 45: $(1, 45), (3, 15), (5, 9)$. Consider their signs since $b = -14$ is negative and $c = 45$ is positive, meaning both numbers in the pair must be negative.

Check which pair of negative factors adds to $-14$: $(-3, -15)$ works because $-3 + (-15) = -14$ and $(-3) \cdot (-15) = 45$.

Rewrite the trinomial as $(x - 3)(x - 15)$. Verify by expanding: $(x - 3)(x - 15) = x^2 - 15x - 3x + 45 = x^2 - 14x + 45$. Thus, the factored form is $(x - 3)(x - 15)$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to 'c' (the constant term) and add to 'b' (the coefficient of the linear term). Understanding this concept is essential for simplifying quadratic expressions and solving equations.

Prime Trinomials

A trinomial is considered prime if it cannot be factored into the product of two binomials with real coefficients. This occurs when the discriminant of the quadratic equation is negative or when no integer pairs exist that satisfy the conditions for factoring. Recognizing prime trinomials is crucial for determining the factorability of quadratic expressions.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation ax^2 + bx + c = 0. While not directly related to factoring, it helps in determining whether a trinomial can be factored by revealing the nature of its roots. If the roots are rational, the trinomial can be factored; if not, it may be prime.

Recommended video:

06:36

Solving Quadratic Equations Using The Quadratic Formula

Related Practice

Textbook Question

Find each product. (x−1)(x+2)

views

Textbook Question

Find the intersection of the sets. {s,e,t}∩{t,e,s}

views

Textbook Question

Evaluate each exponential expression in Exercises 1–22. $2^{3} / 2^{7}$

views

Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $\sqrt{6 x} \cdot \sqrt{3 x^{2}}$

Textbook Question

Multiply or divide as indicated. $\frac{x^3 - 8}{x^2 - 4}\) $\cdot$ \(\frac{x + 2}{3x}$

Textbook Question

Simplify each exponential expression in Exercises 23–64. x⁻²y

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x2−14x+45

Verified video answer for a similar problem:

Key Concepts

Factoring Trinomials

Prime Trinomials

Quadratic Formula

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x²−14x+45