978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Chapter 1 Polynomials 1 This chapter is about polynomials, which include linear and quadratic epressions. When you have completed it, you should be able to add, subtract, multiply and divide polynomials understand the words quotient and remainder used in dividing polynomials be able to use the method of equating coeficients be able to use the remainder theorem and the factor theorem.
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Cambridge International AS and A Level Mathematics: Pure Mathematics & 1.1 Polynomials You already know a good deal about polynomials from your work on quadratics in Chapter 4 of Pure Mathematics 1 (unit P1), because a quadratic is a special case of a polynomial. Here are some eamples of polynomials. + 1 4 1 4 5 1 1 1 7 A (non-zero) polynomial, p() is an epression in of the form n a b n 1 + + + + j + k where a, b, c,..., k are real numbers, a 0, and n is a non-negative integer. n n 1 The number n is called the degree of the polynomial. The epressions a, b,..., j and k which make up the polynomial are called terms. The numbers a, b, c,..., j and k are called coeficients; a is the leading coeficient. The coeficient k is the constant term. Thus, in the quadratic polynomial 4 + 1, the degree is ; the coeficients of and, and the constant term, are 4, and 1 respectively. Polynomials with low degree have special names: if the polynomial has degree 0 it is called a constant polynomial, or a constant degree 1 it is called a linear polynomial degree it is called a quadratic polynomial, or a quadratic degree it is called a cubic polynomial, or a cubic degree 4 it is called a quartic polynomial, or a quartic. When a polynomial is written as a n + b n 1 +... + j + k, with the term of highest degree irst and the other terms in descending degree order inishing with the constant term, the terms are said to be in descending order. If the terms are written in the reverse order, they are said to be in ascending order (or ascending powers of ). For eample, 4 + 7 + 5 is in descending order; in ascending order it is 5 7 + + 4. It is the same polynomial whatever order the terms are written in. 1 The functions 1 1 = and are not polynomials, because the powers of are not positive integers or zero. Polynomials have much in common with integers. You can add them, subtract them and multiply them together and the result is another polynomial. You can even divide a polynomial by another polynomial, as you will see in Section 1.4. 1. Addition, subtraction and multiplication of polynomials To add or subtract two polynomials, you simply add or subtract the coeficients of corresponding powers; in other words, you collect like terms. Suppose that you want to add + 4 to, then you can set out the working like this:
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Chapter 1: Polynomials 4 4 6 Notice that you must leave gaps in places where the coeficient is zero. You need to do addition so often that it is worth getting used to setting out the work in a line, thus: ( 4) ( ) =( 0) +( 1) +( 0 () 1 ) +( ( 4) ( ) ) = + 4 6. You will soon ind that you can miss out the middle step and go straight to the answer. The result of the polynomial calculation ( + 4) ( + 4) is 0. This is a special case, and it is called the zero polynomial. It has no degree. Look back at the deinition of a polynomial, and see why the zero polynomial was not included there. Multiplying polynomials is harder. It relies on the rules for multiplying out brackets: a(b + c +... + k ) = ab + ac +... + ak and (b + c +... + k)a = ba + ca +... + ka. To apply these rules to multiplying the two polynomials 5 + and 5 + 1, replace 5 + 1 for the time being by z. Then ( 5 ) ( 5 + 1) 5 ( 5 + ) = 5 z z = 5 ( 5 1 ) ( 5 1) = ( 10 5 5 ) + ( 6 15 ) = 10 19 10 +. In practice, it is easier to note that every term in the left bracket multiplies every term in the right bracket. You can show this by setting out the steps in the following way. 5 + 1 10 5 + 5 5 10 + 6 15 + ( 5 6) ( ) giving the result 10 19 10 +. 5 15 + It is worth learning to work horizontally. The arrows below show the term 5 from the irst bracket multiplied by 5 from the second bracket to get 5. ( 5 ) ( 5 1) 5 ( 5 + 1) ( 5 + 1) = ( 10 5 + 5 ) = 10 10 +. You could shorten the process and write ( 5 ) ( ) ( 6 15 + ) 5 1 10 5 5 + 6 15 = 10 19 10 +.
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Cambridge International AS and A Level Mathematics: Pure Mathematics & If you multiply a polynomial of degree m by a polynomial of degree n, you have a calculation of the type m ( a b )( ) + b m + (A + B m m + = aa m+ n +... in which the largest power of the product is m + n. Also, the coeficient aa is not zero because neither of a and A is zero. This shows that: When you multiply two polynomials, the degree of the product polynomial is the sum of the degrees of the two polynomials. Eercise 1A 1 State the degree of each of the following polynomials. a + 7 b 5 + 1 c 8 + 5 + 7 + 6 4 d e 5 f 0 4 In each part ind p() + q(), and give your answer in descending order. a p() = + 4 1, q() = + + 7 b p() = 4 + 5 7 +, q() = + 6 c p() = 4 + 7 1, q() = + 5 4 + d p() = + 5, q() = 4 + 5 + 1 e p() = + 4, q() = 1 7 + For each of the pairs of polynomials given in Question ind p() q(). 4 Note that p() + p() may be shortened to p(). Let p() = + 5 and q() = + 4. Epress each of the following as a single polynomial. a p() + q() b p() q() c p() q() d p() q() 5 Find the following polynomial products. a ( )( + 1) b ( + 1)( ) c ( + )( + ) d ( 1)(4 + ) e ( + )( +1) f ( + 1)(4 + 5) g ( + + 6)( + ) h ( + 1)( 5) i (1 + + )( + ) j ( + )(4 5 + ) k ( + 1)( )( + 5) l ( + 1)( )( + 1) 6 In each of the following products find the coefficient of and the coefficient of. a ( + )( + 6) b ( )( + 5) c ( + 1)( 5 + 1) d ( )( + 7) e ( )( 6 + 1) f ( 5)( + 4 + ) g ( + )( + 4) h ( + 1)( 5 + ) i ( + 1)( + + 1) j ( + )(4 5 + 1)
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Chapter 1: Polynomials 7 In each of the following the product of A + B with another polynomial is given. Using the fact that A and B are constants, ind A and B. a (A + B)( ) = 4 11 b (A + B)( + 5) = + 7 15 c (A + B)( ) = 6 d (A + B)( + 5) = 6 + 11 10 e (A + B)( 1) = + f (A + B)( + 4) = + 8 1 g (A + B)( + 4) = 4 + 1 h (A + B)( 1) = 6 7 + 1 1. Equations and identities In this chapter so far you have learned how to add, subtract and multiply polynomials, and you can now carry out calculations such as ( ) +( ) ( 4) ( 1 )( 1++ ) + 1 ( ) 1 5 5 = 1 and fairly automatically. However, you should realise that these are not equations in the normal sense, because they are true for all values of. In P1 Section 10.6, you saw that when two epressions take the same values for every value of the variable, they are said to be identically equal, and a statement such as 1 ( ) ( ) )(1+ 1++ = 1 5 is called an identity. To emphasise that an equation is an identity, the symbol is used. The statement (1 )(1 + + ) 1 means that (1 )(1 + + ) and 1 are equal for all values of. But now suppose that A + B +. What can you say about A and B? As A + B + is an identity, it is true for all values of. In particular, it is true for = 0. Therefore A 0 + B = 0 +, giving B =. But the identity is also true when = 1, so A 1 + = 1 +, giving A =. Therefore: If A + B +, then A = and B =. This is an eample of the process called equating coeficients. The full result is: If a n + b n 1 +... + k A n + B n 1 +... + K, then a = A, b = B,..., k = K. The statement in the bo says that, if two polynomials are equal for all values of, then all the coeficients of corresponding powers of are equal. This result may not surprise you, but you should be aware that you are using it. Indeed, it is very likely that you have used it before now without being aware of it.
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Cambridge International AS and A Level Mathematics: Pure Mathematics & EXAMPLE 1..1 One factor of 5 is. Find the other factor. There is nothing wrong in writing down the answer by inspection as + 1. But the process behind this quick solution is as follows. Suppose that the other factor is A + B. Then (A + B)( ) 5, and, multiplying out, you get +( ) A AA B B B 5. By equating coeficients of, you get A =. Equating coeficients of 0, the constant term, you get B =, giving B = 1. Therefore the other factor is + 1. You can also check that the middle term, A + B = 6 + 1 = 5, is correct. You should continue to write down the other factor by inspection if you can. However, in some cases, it is not easy to see what the answer will be without intermediate working. EXAMPLE 1.. 6 If 4 + + ( )(A + B + C) + R, ind A, B, C and R. Multiplying out the right side gives 4 + + A + ( A + B) + ( B + C) + ( C + R). Equating coeficients of : 4 = A. Equating coeficients of : = A + B = 4 + B = 8 + B, so B = 10. Equating coeficients of : 0 = B + C = 0 + C, so C = 0. Equating coeficients of 0 : = C + R = 40 + R, giving R = 4. Therefore A = 4, B = 10, C = 0 and R = 4, so 4 + + ( )(4 + 10 + 0) + 4. In practice, people often use the symbol for equality, =, when they really mean the symbol for identity,. The contet usually suggests which meaning is intended. Eercise 1B 1 In each of the following quadratic polynomials one factor is given. Find the other factor. a + 1 ( + 4)( ) b + 14 51 ( )( ) c + 5 ( )( ) d 5 + 48 7 (5 + 9)( ) e 15 ( + 5)( ) f 14 + 1 10 ( + 5)( ) In each of the following identities ind the values of A, B and R. a + 7 ( + )(A + B) + R b + 9 ( + 1)(A + B) + R
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Chapter 1: Polynomials c 15 14 8 (5 + )(A + B) + R d 6 + 5 ( + 1)(A + B) + R e 1 5 + ( )(A + B) + R f 1 11 + 6 ( )(A + B) + R In each of the following identities ind the values of A, B, C and R. a + 1 ( + )(A + B + C) + R b 5 + 10 + 10 ( )(A + B + C) + R c + + 4 ( 1)(A + B + C) + R d 1 + 11 7 + 5 ( + )(A + B + C) + R e 4 + 4 7 + 5 ( 5)(A + B + C) + R f 9 + 1 15 10 ( + 4)(A + B + C) + R 4 In each of the following identities ind the values of A, B, C, D and R. a 4 + 5 + 11 5 ( + )(A + B + C + D) + R b 4 4 7 + 7 ( )(A + B + C + D) + R c 6 4 + 5 + + ( + 1)(A + B + C + D) + R d 4 7 + 17 14 + 5 ( 1)(A + B + C + D) + R 1.4 Division of polynomials You can, if you wish, carry out division of polynomials using a layout like the one for long division of integers. You might already have seen and used such a process, and eamples are given below. However, you can also use the method of equating coeficients for division, as will now be eplained. When you divide 11 by 9, you get an answer of 1 with 4 over. The number 9 is called the divisor, 1 is the quotient and 4 the remainder. You can epress this as an equation in integers, 11 = 9 1 + 4. The remainder r has to satisfy the inequality 0 r < 9. Now look back at Eample 1... You will see that it is an identity of just the same shape, but with polynomials instead of integers. So you can say that, when 4 + + is divided by the divisor, the quotient is 4 + 10 + 0 and the remainder is 4. The degree of the remainder (in this case 0) has to be less than the degree of the divisor. The degree of the quotient 4 + 10 + 0, which is, is equal to the difference between the degree of the polynomial 4 + +, which is, and the degree of the divisor, which is 1. 7 When a polynomial, a(), is divided by a non-constant divisor, b(), the quotient q() and the remainder r() are deined by the identity a() b()q() + r(), where the degree of the remainder is less than the degree of the divisor. The degree of the quotient is equal to the degree of a() the degree of b().
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Cambridge International AS and A Level Mathematics: Pure Mathematics & EXAMPLE 1.4.1 Find the quotient and remainder when 4 + + is divided by + 1. Equating coeficients approach Using the result in the bo, as the degree of 4 + + is 4 and the degree of + 1 is 1, the degree of the quotient is 4 1 =. And as the degree of the remainder is less than 1, the remainder is a constant. Let the quotient be A + B + C + D, and let the remainder be R. Then 8 so 4 ++ ( 1) ( A B C D ) + R, 4 + + A 4 + (A + B) + (B + C) + (C + D) + D + R. Equating coeficients of 4 : 1 = A. Equating coeficients of : 0 = A + B, so B A, giving B = 1. Equating coeficients of : 0 = B + C, so C = B, giving C = 1. Equating coeficients of : 1 = C + D, so D = 1 C, giving D = 0. Equating coeficients of 0 : = D + R, so R = D, giving R =. The quotient is + and the remainder is. Long division approach To divide 4 + + by + 1, you start by laying out the division as with integers. It is wise to indicate the absence of any or terms by writing 0 and 0 as placeholders, just as you use zeros to indicate the absence of any hundreds or tens in the number 4007. So you have: 4 +1 ) +0 +0 + + Now note that the (the leading term of + 1) divides into the leading term of 4 + +, namely 4, eactly times, so you get: 4 +1 ) +0 +0 + + + 4 You continue the division in this way: bringing down the net term, 0, and noting that divides into eactly times, you get: 4 +1 ) +0 +0 + + + 4 + 0
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Chapter 1: Polynomials Continuing in this way, the inal result is + 4 +1 ) +0 +0 + + + 4 + 0 + + ( 0) So the quotient is + and the remainder is. EXAMPLE 1.4. Find the quotient and remainder when 4 + is divided by +. Equating coeficients approach The result in the bo states that the degree of the remainder is less than, so assume that it is a linear polynomial. Let the quotient be A + B + C, and the remainder be R + S. Then 4 + ( + )(A + B + C) + R + S, so 4 + A 4 + ( A + B) + (A B + C) Equating coeficients of 4 : 1 = A. + (B C + R) + C + S. Equating coeficients of : 0 = A + B, so B = A, giving B =. Equating coeficients of : = A B + C, so C = A + B, giving C = 5. Equating coeficients of : 0 = B C + R, so R = B + C, giving R = 6. Equating coeficients of 0 : = C + S, so S = C, giving S = 1. The quotient is + + 5 and the remainder is 6 1. Long division approach When dividing 4 + by +, the leading term of + is, so the calculation goes as follows: 9 + + 5 + ) 4 + 0 + + 0 4 ) + 4 + + 0 + 0 4 + 4 5 4 5 10 + 10 6 1 So again, it is found that the quotient is + + 5 and the remainder is 6 1.
978-1-16-600-1 Cambridge International AS and A Level Mathematics: Pure Mathematics and Revised Edition Coursebook Ecerpt Cambridge International AS and A Level Mathematics: Pure Mathematics & When you are dividing by a linear polynomial, there is a quick way of inding the remainder. For eample, in Eample 1.4.1, when 4 + + was divided by + 1, the irst line of the solution was: 4 + + ( + 1)(A + B + C + D) + R. Since this is an identity, it is true for all values of and, in particular, it is true for = 1. Putting = 1 in the left side, you get ( 1) 4 + ( 1) + = ; putting = 1 in the right side, you get 0 (A( 1) + B( 1) + C( 1) + D) + R, which is simply R. Therefore R =. Similar reasoning leads to the remainder theorem. Remainder theorem When a polynomial p() is divided by t, the remainder is the constant p(t). Proof When p() is divided by t, let the quotient be q() and the remainder be R. Then p() ( t)q() + R. Putting = t in this identity gives p(t) = 0 q(t) + R = R, so R = p(t). 10 EXAMPLE 1.4. Find the remainder when + 4 is divided by +. Let p() = + 4. Then p( ) = ( ) ( ) + 4 = 7 + 9 + 4 = 14. By the remainder theorem, the remainder is 14. EXAMPLE 1.4.4 When the polynomial p() + a + b is divided by 1 the remainder is 4. When p() is divided by the remainder is also 4. Find the remainder when p() is divided by. By the remainder theorem, when p() is divided by 1, the remainder is p(1) = 1 1 + a + b = a + b. Therefore a + b = 4, so a + b =. Similarly, p() = + a + b = a + b 4, so a + b 4 = 4 and a + b = 0. Solving the equations a + b = and a + b = 0 simultaneously gives a = and b = 4, making the polynomial p() = + 4. The remainder on division by is p() = + 4 =. The remainder theorem is useful for inding the remainder when you divide a polynomial by a linear polynomial such as, but it doesn t tell you how to ind the remainder when you divide by a linear polynomial such as. To do this, you need the etended form of the remainder theorem.