Mathematics
Factorising expressions
Factorising expressions involves breaking down an algebraic expression into a product of its factors. This process helps to simplify and solve equations, as well as identify common factors within an expression. By factoring, one can often find solutions to equations and gain a deeper understanding of the relationships between different terms in the expression.
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8 Key excerpts on "Factorising expressions"
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
Fractions reduce more easily, equations solve more easily, and answers are observed more easily when you can factor. 180 Part II: Algebra is Part of Everything Factoring out numbers Factoring is the opposite of distributing; it’s ‘undistributing’ (refer to Chapter 7 for more on distribution). When performing distribution, you multiply a series of terms by a common multiplier. Now, by factoring, you seek to find what a series of terms has in common and then take it away, dividing the common factor or multiplier out from each term. Think of each term as a numerator of a fraction, and you’re finding the same denominator for each. By factoring out, the common factor is put outside parentheses or brackets and all the results of the divisions are left inside. An expression can be written as the product of the largest number that divides all the terms evenly times the results of the divisions: ab + ac + ad = a(b + c + d ). Writing factoring as division In the trinomial 16a − 8b + 40c 2 , 2 is a common factor. But 4 is also a common factor, and 8 is a common factor. Here are the divisions of the terms by 2, 4 and 8: Reviewing the terms and rules You’ll understand factoring better if you have a firm handle on what the terms used to talk about factoring mean: 6 Term: A group of number(s) and/or varia- ble(s) connected to one another by multi- plication or division and separated from other terms by addition or subtraction. 6 Factor: Any of the values involved in a multiplication problem that, when multi- plied together, produce a result. 6 Coefficient: A number that multiplies a variable and tells how many of the variable. 6 Constant: A number or variable that never changes in value. 6 Relatively prime: Terms that have no factors in common. If the only factor that numbers share in common is 1, they’re considered relatively prime. Here is an illustration for all the terms I just gave: In the expression 5xy + 4z − 6, you see three terms.
eBook - ePub- Toby Wagner(Author)
- 2021(Publication Date)
- Chemeketa Press(Publisher)
CHAPTER 5Factoring
When multiplying, the result is known as a product, and the numbers being multiplied are known as factors. In the equation 3 – 2 = 6, for example, the product is 6, and the factors are 3 and 2.Factoringis the process of breaking a number or algebraic expression down into its factors. In previous math courses, you may have used this process to create prime factorizations. For example, the prime factorization of 24 is 2 · 2 · 2 · 3. In this chapter, we will be factoring polynomials.Our primary focus will be on second-degree polynomials, but we will also encounter higher-degree expressions. After learning the various methods and techniques that can be used when factoring polynomials, we’ll use factoring to help us simplify rational expressions.This chapter is composed of the following sections:5.2 Factoring Trinomials of the Form x2 + bx + c5.1 Factoring out the Greatest Common Factor5.3 Special Cases for Factoring5.4 Simplifying Rational Expressions5.1 Factoring Out the Greatest Common Factor
Overview
At this point, you should be comfortable with operations — addition, subtraction, multiplication, division, and exponents — and how they pertain to polynomials. Now it’s time to learn how to reverse the multiplication process. Here’s a problem that does this:Factor out the GCF: 9x3 + 18x2 + 27xGCF stands for Greatest Common Factor, but what does the GCF look like when variables are involved? And how does one go about factoring out a GCF? We need answers for these questions before we are ready to tackle this problem.This section will introduce you to factoring and teach you how to: ◆ Determine the other factor when given a polynomial and one monomial factor ◆ Identify and factor out the greatest common factor (GCF) ◆ Use grouping to factor polynomialsA. Finding a Missing Factor
In Chapter 4
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
326 CHAPTER 10 ◆ Factoring Algebraic Expressions Finding Monomial Factors 10.1 10.1 Factoring an algebraic expression , like finding the prime factors of a number, means writing the expression as a product of factors . The prime factorization of 12 is 2 ? 2 ? 3. Other factorizations of 12 are 2 ? 6 and 4 ? 3 . Since factorization means writing a number or an algebraic expression as a product, then a number or an algebraic expression divided by one factor generates another factor. Thus, 12 divided by 2 gives 6, so 2 times 6 is a fac-torization of 12. To factor the expression 2 x 1 2 y , notice that 2 is a factor common to both terms of the expression. In other words, 2 is a factor of 2 x 1 2 y . To find the other factor, divide by 2. 2 x 1 2 y 2 5 2 x 2 1 2 y 2 5 x 1 y Therefore, a factorization of 2 x 1 2 y is 2( x 1 y ). A monomial factor is a one-term factor that divides each term of an algebraic expres-sion. Here, 2 divides each term of the algebraic expression and is called a monomial factor. When factoring any algebraic expression, always look first for monomial factors that are common to all terms. Factor: 3 a 1 6 b . First, look for a common monomial factor. Since 3 divides both 3 a and 6 b , 3 is a common monomial factor of 3 a 1 6 b . Divide 3 a 1 6 b by 3. 3 a 1 6 b 3 5 3 a 3 1 6 b 3 5 a 1 2 b Thus, 3 a 1 6 b 5 3( a 1 2 b ). Check this result by multiplication: 3( a 1 2 b ) 5 3 a 1 6 b . ◆ Factor: 4 x 2 1 8 x 1 12. Since 4 divides each term of the expression, divide 4 x 2 1 8 x 1 12 by 4 to obtain the other factor. 4 x 2 1 8 x 1 12 4 5 4 x 2 4 1 8 x 4 1 12 4 5 x 2 1 2 x 1 3 Thus, 4 x 2 1 8 x 1 12 5 4( x 2 1 2 x 1 3). 4( x 2 1 2 x 1 3) 5 4 x 2 1 8 x 1 12 Your product should be the original expression. NOTE: In this example, 2 is also a common factor of each term of the expression. However, 4 is the greatest common factor. The greatest common factor of a polynomial is the largest common factor that divides all terms in the expression.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
89 N1 Mathematics| Hands-On Factorising by grouping Expression with more than three terms can be factorised by grouping the terms in order to find a common factor. In some expressions a common factor is not always obvious. Sometimes grouping or rearranging the terms to obtain a much simpler expression is necessary before factorisation. Example 1 Factorise the expression ax + ay + bx + by . Solution ax + ay + bx + by • Four terms. There is no common factor in the given expression. = ( ax + ay ) + ( bx + by ) • Use brackets and group the terms to take a common factor out in each bracket. Put a positive sign between the brackets. = a ( x + y ) + b ( x + y ) • a is a common factor of the first bracket and b is a common factor of the second bracket. This expression consists of a sum of two terms. = ( x + y )( a + b ) • Each term has a common factor of ( x + y ), so it can be factorised further. This fully factorised expression consists of one term. Check the answer. ( x + y )( a + b ) • Use FOIL. = ax + bx + ay + by • This is equal to ax + ay + bx + by , the given expression. Example 2 Factorise the expression ab – 6 xy + 2 bx – 3 ay . Solution ab – 6 xy + 2 bx – 3 ay • Four terms. There is no common factor in the given expression. = ab + 2 bx – 6 xy – 3 ay • Rearrange the terms. = ( ab + 2 bx ) + (–6 xy – 3 ay ) • Use brackets and group the terms so that a common factor can be taken out in each bracket. Put a positive sign between the brackets. 90 Module 3 • Factorisation and fractions = b ( a + 2 x ) + 3 y (–2 x – a ) • b is a common factor of the first bracket and 3 y is a common factor of the second bracket. This expression consists of a sum of two terms. = b ( a + 2 x ) – 3 y (2 x + a ) • Change the signs of the second bracket to make a common bracket that can be taken out as a common factor. = b ( a + 2 x ) – 3 y ( a + 2 x ) • Note: 2 x + a = a + 2 x = ( a + 2 x )( b – 3 y ) • Each term has a common factor of ( a + 2 x ), so it can be factorised further.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
3 MODULE Factorisation and fractions 3.1 Factorisation, HCF and LCM On completion of this module, you should be able to: 3.1.1 Find the factors of an expression 3.1.2 Determine the HCF (highest common factor) 3.1.3 Determine the LCM (lowest common multiple) 3.1.4 Factorise polynomials with common factors, and regroup terms with a common factor. 3.2 Algebraic fractions On completion of this module, you should be able to: 3.2.1 Simplify fractions 3.2.2 Multiply and divide fractions 3.2.3 Add and subtract algebraic fractions. 76 Module 3 • Factorisation and fractions 3.1 Factorisation, HCF and LCM Introduction • The process of writing a mathematical statement as a product of its factors is called factorisation . • Factorisation of expressions is the reverse process of developing products (expanding). • The factors of an expression are two or more expressions that can be multiplied to produce the original expression. Example ax + bx + cx = x ( a + b + c ) • x and ( a + b + c ) are factors of all three terms, ax + bx + cx . By expanding: x ( a + b + c ) = ax + bx + cx Consider the following. multiplication x ( a + b + c ) = ax + bx + cx factorisation Pre-knowledge • A monomial is a single algebraic term, for example 3 x 2 y 3 z or 2( x + y ). • Polynomials are algebraic expressions with two or more terms that involve numbers and variables (symbols such as x , y , z ). The variables must have positive integer exponents and can only appear in the numerator of the expression. Example Consider the polynomial: 2 x 3 – 4 x 2 + 3 x – 1 The degree of the expression is 3 (highest power of x in the expression). The coefficient of x 2 is –4. The exponent (index) of the variable in the third term is 1. The constant term is –1. The variable in the expression is x. There are 4 terms. The polynomial is written in descending powers of x. • 2 x 3 – 4 x 2 + 3 x – 1 is a polynomial, but 2 8 x + 12 x 6 – x 1 2 is not a polynomial.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
The distance x to the center of gravity is given by Could you solve this equation for, say, ? In this chapter we show how to solve this type of equation. m 1 x 10m 1 25m 2 m 1 m 2 320 Chapter 11 ◆ Factoring and Fractions Here we will show traditional pencil-and-paper methods of solution alongside solutions by a calculator such as the TI-89 that can manipulate algebraic expressions. 11–1 Common Factors As usual, we must start with definitions of the terms we will be using in the following sections. Factors of an Expression The factors of an expression are those quantities whose product is the original expression. ◆◆◆ Example 1: The factors of are and , because ◆◆◆ Many expressions have no factors other than 1 and themselves. Such expres- sions are called prime. Factoring is the process of finding the factors of an expression. It is the reverse of finding the product of two or more quantities. We usually factor an expression by recognizing the form of that expression. In the first type of factoring we will cover, we look for common factors. Common Factors ■ Exploration: Try this. Expand this expression by multiplying out. Now how would you return the expression you just got back to its original form? Can you state your findings as a general rule? ■ If each term of an expression contains the same quantity (called the common factor), the quantity may be factored out. Common Factor 10 This is nothing but the distributive law that we studied earlier. ◆◆◆ Example 2: In the expression x 3 3x ab ac a(b c) a(b c d) x 2 4x x(x 4) Factoring (Finding the Factors) x(x 4) x 2 4x Multiplication (Finding the Product) x 2 9 (x 3)(x 3) x 2 3x 3x 9 x 3 x 3 x 2 9 Section 1 ◆ Common Factors 321 each term contains an x as a common factor. So we write Most of the factoring we will do will be of this type. ◆◆◆ ◆◆◆ Example 3: Here are some examples of the factoring out of common factors.
eBook - PDFMathematics NQF3 SB
TVET FIRST
- M Van Rensburg, I Mapaling, M Trollope A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
4 − x 2 and x 2 − 2 x 4.1.2 Factorising quadratic expressions Quadratic relates to a second degree expression where the highest exponent is 2. We can factorise a quadratic expression that contains two terms by taking out the HCF, as shown in Example 4.3. Tip: Remember that we can rewrite an equation with parentheses so that the solution is still the same: b − a = − ( a − b ). Example 4.3 Factorise: 8 x 2 − 24 x. Solution: First find the HCF: 8 x 2 = 2 3 × x × x [Write the first term as a multiple of its smallest or prime factors ] 24 x = 2 3 × 3 × x [Write the second term as a multiple of its smallest or prime factors] HCF = 2 3 × x = 8 x [Decide which factors are common between the first and second factors; this will be your HCF] 2 3 and x are common in both terms therefore the common factor will include both these terms as a product of each other. Now go back to the original expression and use the HCF to factorise. Method 1: Take out the common factor in the original calculation: 8 x 2 − 24 x = 8 x ( x − 3) [8 x is the HCF] or quadratic: an expression containing a term that is squared; an expression with the highest degree of 2 prime factors: a factor that only has 1 and itself as factors, no other factors can divide into a prime factor New words 114 Module 4 Example 4.3 continued Method 2: 8 x 2 − 24 x [The HCF is 8 x , therefore take out the common factor and write it outside the bracket] = 8 x ( 8 x 2 ___ 8 x − 24 x ____ 8 x ) [The terms in the bracket is the expression divided by the HCF 8 x ] = 8 x ( x − 3 ) [Cancel like factors] A quadratic trinomial equation is written in the standard form ax 2 + bx + c = 0, where a ≠ 0. This trinomial has three types of terms: l The term ax 2 is the quadratic term because it contains a square ( x 2 ). l The term bx is the linear term because its exponent is 1 ( x 1 ). l The term c is the constant term with an exponent of zero ( x 0 ).
eBook - PDFMathematics NQF4 SB
TVET FIRST
- M Van Rensburg, I Mapaling M Trollope(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
53 Module 3 Work with algebraic expressions using the remainder and factor theorems Module 3 Overview By the end of this module you should be able to: • Unit 3.1: Use and apply the remainder theorem and the factor theorem to: – Find the remainder. – Prove that an expression is a factor. – Find an unknown variable to make an expression a factor or to leave a remainder. • Unit 3.2: Factorise third-degree polynomials including examples that require the factor theorem. Unit 3.1: Use and apply the remainder theorem and the factor theorem 3.1.1 Polynomials Topic 2: Functions and algebra Polynomials are mostly used to approximate unknown functions. One method of doing this, is to divide a polynomial into factors. In this module, we will discuss the division of polynomials, first by using long division, and then by using the remainder theorem. We also explain the factor theorem, which is used to find factors of a polynomial. A polynomial is an algebraic expression that contains two or more terms, with each term consisting of a number (called the coefficient ) and a variable raised to a positive power (counting number). The math symbols (operators) + and – separate the terms. The term with the highest exponent is written first in a polynomial. For example, 2 a 3 + 4 a 2 – a + 9 is a polynomial with four terms. Here, the first term, 2 a 3 , is a ‘third-degree’ term. The second term, 4 a 2 , is a ‘second-degree’ term and the third term, a , is a ‘first-degree’ term. (This is because a = a 1 ). The fourth term, 9, contains no variable such as x , y or a , and is the constant. The degree of the polynomial is given by the exponent of the highest power. Therefore, 2 a 3 + 4 a 2 – a + 9 is a third-degree polynomial . Later in this module you will learn how to factorise third-degree polynomials. Here are different types of polynomials: • In the form ax + b , for example 2 x + 3: this is a linear expression, or polynomial of degree 1.
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