/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Factor completely. If the polyno... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 26

Factor completely. If the polynomial cannot be factored, write prime. \(a^{2}+9 a+20\)

Short Answer

Expert verified
\(a^2 + 9a + 20 = (a + 4)(a + 5)\)

Step by step solution

01

- Identify the coefficients

Recognize the quadratic polynomial in the form of \[ a^2 + 9a + 20 \]. The coefficients from this polynomial are: - Coefficient of \(a^2\): 1 - Coefficient of \(a\): 9 - Constant term: 20
02

- Find two numbers that multiply to 20 and add to 9

Find two numbers whose product equals the constant term (20) and whose sum equals the coefficient of the linear term (9). These numbers are 4 and 5 because: \[ 4 \times 5 = 20 \] \[ 4 + 5 = 9 \]
03

- Write the polynomial as a product of binomials

Express the quadratic polynomial as a product of binomials using the numbers found in Step 2: \[ a^2 + 9a + 20 = (a + 4)(a + 5) \]
04

- Verify the factorization

To ensure the factorization is correct, expand \((a + 4)(a + 5)\): \[(a + 4)(a + 5) = a(a + 5) + 4(a + 5) \] \[= a^2 + 5a + 4a + 20 \] \[= a^2 + 9a + 20 \]. The factorization is verified as accurate.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler components, often as a product of other polynomials. For instance, the polynomial \( a^2 + 9a + 20 \) can be factored into \( (a + 4)(a + 5) \). This means it is expressed as the product of two binomials. Factoring helps in simplifying polynomial expressions and solving polynomial equations. Here's a step-by-step guide for polynomial factorization:
  • Identify the coefficients and constant term.
  • Find pairs of numbers (factors) that multiply to the constant term and add to the middle coefficient.
  • Rewrite the polynomial as a product of binomials or other polynomials.
  • Verify by expanding the factors back to the original polynomial.
Mastering these steps will make polynomial factorization much simpler.
Quadratic Equations
Quadratic equations are polynomial equations of degree two, typically written in the form \( ax^2 + bx + c = 0 \). They are fundamental in algebra and come up often in various math problems. Here, we work with the quadratic polynomial \( a^2 + 9a + 20 \). To factorize it:
  • Recognize the standard quadratic form and identify coefficients.
  • Use these coefficients to find appropriate factors.
  • Rewrite the quadratic using these factors as binomial products.
Understanding quadratic equations helps in recognizing patterns that make factorization easier and solving for variable values.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). For instance, \( a^2 + 9a + 20 \) is an algebraic expression. Handling these expressions involves several skills:
  • Simplifying by factoring or combining like terms.
  • Expanding expressions using distributive property.
  • Solving equations derived from expressions.
In our example, factoring helps break down the expression into simpler parts, making it easier to work with. Practice with various algebraic expressions improves problem-solving abilities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor completely. \(m^{3} n-10 m^{2} n^{2}+24 m n^{3}\)

Factor each trinomial completely. $$ 4 x^{4}+2 x^{3}+x^{2} $$

Factor each trinomial completely. See Examples 1鈥�7. ( Hint: In Exercises 55鈥�58, first write the trinomial in descending powers and then factor.) $$ 24 x^{4}+55 x^{2}-24 $$

Factor each polynomial completely. $$ (m+n)^{2}-(m-n)^{2} $$

Factor completely. \(12 k^{5}-6 k^{3}+10 k^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.