Chapter 7 More About Polynomials

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7.1 Polynomials and Their Operations 7.2 Division of Polynomials 7.3 Remainder Theorem Home

7.4 Factor Theorem and Its Applications

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7.1 Polynomials and Their Operations A. Polynomials in One Variable In general, a polynomial in one variable x can be expressed as an x n + an −1 x n −1 + ... + a1 x + a0 , where n is a non - negative integer and an , an −1, , a1 , a0are real numbers with an ≠ 0. Also , n is called the degree of the polynomial . Remarks: 1 1 and are not polynomial s. x x +1

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Expression such as x − 2 , x , x +

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A polynomial in one variable x can be regarded as a function of x.

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A polynomial with a non-zero constant term a0only is of degree 0. A polynomial consisting of 0 only is called the zero polynomial and has no degree.

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7.1 Polynomials and Their Operations B. Addition, Subtraction and Multiplication of Polynomials Example 7.1T Add x 3 + 7 x 2 − 2 and 1 + 2 x + 3 x 2 + 4 x 3 . Solution: ( x 3 + 7 x 2 − 2) + (1 + 2 x + 3 x 2 + 4 x 3 ) = x 3 + 4 x 3 + 7 x 2 + 3x 2 + 2 x − 2 + 1 Home

= 5 x 3 + 10 x 2 + 2 x − 1

Group the like terms to calculate the coefficients.

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7.1 Polynomials and Their Operations Example 7.2T Subtract x 3 + 7 x 2 − 2 from 1 + 2 x + 3 x 2 + 4 x 3 . Solution:

Subtract A from B means B – A.

(1 + 2 x + 3 x 2 + 4 x 3 ) − ( x 3 + 7 x 2 − 2) = 1 + 2 x + 3x 2 + 4 x 3 − x 3 − 7 x 2 + 2 = 4 x3 − x3 + 3x 2 − 7 x 2 + 2 x + 1 + 2 = 3x3 − 4 x 3 + 2 x + 3 Home Content

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7.1 Polynomials and Their Operations C. Equality of Polynomials Two polynomials f (x) and g(x) are equal for all real values of x if the coefficients of like powers of x in the two polynomials are equal. Also, we can write f ( x) ≡ g ( x) to denote two identical polynomial s f ( x) and g ( x).

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f (x) ≡ g(x) means that f (x) and g(x) are equal for all real values of x. Also, we can say that f (x) is identical to g(x), or f (x) ≡ g(x) is an identity.

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7.2 Division of Polynomials A. By Monomials For example, we divide 2 x 2 + 3 x − 4 by − x. The polynomial 2 x 2 + 3 x − 4 is called the dividend and the polynomial − x is called the divisor. When a polynomial is divided by a divisor, we get a quotient and a remainder. We can write dividend = divisor × quotient + remainder Home Content

where the remainder is zero or the degree of the remainder is less than that of the divisor. It is known as the Division algorithm of polynomials.

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7.2 Division of Polynomials B. By Linear Binomials Consider 2 x 2 + 3 x − 4 divided by x − 2. Step 1:

2x x − 2 2 x 2 + 3x − 4 2x2

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Step 2:

2x x − 2 2 x 2 + 3x − 4 2x2 − 4x 7x − 4

The first term 2 x in the quotient is obtained by dividing 2 x 2 (in the dividend) 2x2 by x (in the divisor), that is, = 2 x. x

2x2 – 4x is obtained by multiplying x – 2 and 2x. Then 2x2 – 4x is subtracted from the dividend 2x2 + 3x – 4 to give 7x – 4.

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7.2 Division of Polynomials Step 3:

2x + 7 x − 2 2 x 2 + 3x − 4 2x2 − 4x 7x − 4 7x

Step 4:

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2x + 7 x − 2 2 x 2 + 3x − 4 2x2 − 4x 7x − 4 7 x − 14 10

The second term + 7 in the quotient is obtained by dividing 7 x by x (in the 7x divisor), that is, = 7. x

7x – 14 is obtained by multiplying x – 2 by 7. Then 7x – 14 is subtracted from 7x – 4 and the remainder is 10. We stop here since the degree of the remainder is less than the degree of the divisor.

We can also write 2 x 2 + 3 x − 4 = ( x − 2)(2 x + 7) + 10.

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7.2 Division of Polynomials C. By Quadratic Polynomials Example 7.7T Find the quotient and the remainder when 2 x 3 + 11x − 3 is divided by 1 + 3x + x 2 . Solution: 2x − 6

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x 2 + 3 x +1 2 x 3 + 0 x 2 +11x − 3 2 x3 + 6 x 2 + 2 x − 6x2 + 9x − 3 − 6 x 2 − 18 x − 6 27 x + 3 ∴ quotient = 2 x − 6 remainder = 27 x + 3

• To start with the method of long division, we need to rearrange the terms of the dividend and the divisor in descending powers of x. • The procedure stops when the degree of the remainder is less than the degree of the divisor.

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7.2 Division of Polynomials Example 7.8T Find the quotient and the remainder when 2 x 3 + 7 x 2 + 7 x + 5 is divided by 1 + x + x2. Solution:

2x + 5

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x 2 + x +1 2 x 3 + 7 x 2 + 7 x + 5 2 x3 + 2 x 2 + 2 x 5x2 + 5x + 5 5x2 + 5x + 5 0

Since the remainder is 0, the polynomial 2x3 + 7x2 + 7x + 5 is divisible by 1 + x + x2.

∴ Quotient = 2 x + 5 Remainder = 0

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7.3 Remainder Theorem When a polynomial P(x) is divided by x – a, a quotient Q(x) and a remainder R is obtained. P(x) can be expressed in terms of x – a, Q(x) and R as P( x) = ( x − a ) × Q( x) + R.....................(*) By putting x = a into (*), we have P(a) = (a − a ) × Q(a ) + R = 0 × Q(a ) + R = 0+ R =R ∴ R = P(a ) Home Content

Remainder Theorem When a polynomial P(x) is divided by x – a, the remainder R is equal to P(a).

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7.3 Remainder Theorem When a polynomial P(x) is divided by px – q, the quotient Q(x) and the remainder R are obtained. Then we have P ( x) = ( px − q ) × Q( x) + R.....................(**) Similarly, (**) is also an identity. By putting x =

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q into (**), we have p

q  q  q P  =  p  − q  × Q  + R  p   p   p q = 0 × Q  + R  p =0+ R =R

q ∴ R = P   p

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7.4 Factor Theorem and Its Applications A. Factor Theorem Factor Theorem If P(x) is a polynomial and P(a) = 0, then x – a is a factor of P(x). Conversely, if x – a is a factor of a polynomial P(x), then P(a) = 0.

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q If P(x) is a polynomial and P  = 0, then px – q is a factor of P(x).  p q Conversely, if px – q is a factor of a polynomial P(x), then P  = 0.  p

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7.4 Factor Theorem and Its Applications B. Applications of the Factor Theorem Example 7.12T Let P( x) = x 3 + 2 x 2 − 3 x − 4. •

Factorize P(x).

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Solve the equation P(x) = 0.

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(a)

P( −1) = ( −1)3 + 2 × ( −1) 2 − 3 × (−1) − 4 = −1 + 2 + 3 − 4 =0 ∴ x + 1 is a factor of P(x). By the method of long division, P ( x) = ( x + 1)( x 2 + x − 4).

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7.4 Factor Theorem and Its Applications Example 7.12T Let P( x) = x 3 + 2 x 2 − 3 x − 4. •

Factorize P(x).

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Solve the equation P(x) = 0.

Solution: (b) P ( x) = 0, ( x + 1)( x 2 + x − 4) = 0 Home Content

x + 1 = 0 or x 2 + x − 4 = 0 x = −1

− 1 ± 12 − 4 × 1 × (−4) or x = 2 ×1 − 1 ± 17 = 2

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