Chapter 6 More about Polynomials

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Presentation transcript:

Chapter 6 More about Polynomials Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8 Example 9 Example 10 Additional Example 6.1 Additional Example 6.2 Additional Example 6.3 Additional Example 6.4 Additional Example 6.5 Additional Example 6.6 Additional Example 6.7 Additional Example 6.8 Additional Example 6.9 Additional Example 6.10 Quit New Trend Mathematics - S4A

Chapter 6 More about Polynomials Example 11 Example 12 Example 13 Example 14 Example 15 Example 16 Example 17 Additional Example 6.11 Additional Example 6.12 Additional Example 6.13 Additional Example 6.14 Additional Example 6.15 Additional Example 6.16 Additional Example 6.17 Quit New Trend Mathematics - S4A

Additional Example

Simplify the following. Additional Example 6.1 Simplify the following. Solution:

Additional Example 6.1 Solution:

Additional Example

Simplify the following. Additional Example 6.2 Simplify the following. Solution:

Additional Example 6.2 Solution:

Additional Example

Additional Example 6.3 Expand the following. Solution:

Additional Example 6.3 Solution:

Additional Example

Additional Example 6.4 Find the quotient and the remainder of each of the following divisions. Solution: (a)

Additional Example 6.4 (b) Solution:

Additional Example

Additional Example 6.5 Find the quotient and the remainder of each of the following divisions. Solution: (a)

Additional Example 6.5 (b) Solution:

Additional Example

Additional Example 6.6 Find the quotient and the remainder of each of the following divisions. (a) Solution:

Additional Example 6.6 (b) Solution:

Additional Example

Additional Example 6.7 When is divided by , the quotient is x + 6 and the remainder is 3x  24. Find the values of p and q. Solution:

Additional Example

Additional Example

Find the remainder when is divided by Additional Example 6.8 Find the remainder when is divided by Solution: (a) By the remainder theorem,

(b) By the remainder theorem, Solution: Additional Example 6.8 (b) By the remainder theorem, Solution: (c) By the remainder theorem,

Additional Example

When is divided by , the remainder is 1. Find the value of k. Additional Example 6.9 When is divided by , the remainder is 1. Find the value of k. Solution: By the remainder theorem,

Additional Example

Additional Example

When f (x) is divided by x + 3, the remainder is -170. Solution: Additional Example 6.10 When is divided by x + 3 and 2x  1, the remainders are 170 and 16 respectively. Find the values of p and q. When f (x) is divided by x + 3, the remainder is -170. Solution:

When f (x) is divided by 2x - 1, the remainder is -16. Additional Example 6.10 When f (x) is divided by 2x - 1, the remainder is -16. Substitute p = -8 into (2), Solution:

Additional Example

Additional Example

Additional Example 6.11 Let . Use the factor theorem to determine whether each of the following is a factor of h(x). (a) x  1 (b) x + 2 (c) 2x + 3 (d) 3x  2 Solution:

Additional Example 6.11 Solution:

Additional Example

Additional Example

Consider the polynomial (a) Prove that 2x  5 is a factor of g(x). Additional Example 6.12 Consider the polynomial (a) Prove that 2x  5 is a factor of g(x). (b) Factorize g(x). (c) Solve the equation g(x) = 0. Solution:

Additional Example 6.12 (b) By long division, Solution:

Additional Example 6.12 Solution:

Additional Example

Additional Example

The polynomial is divisible by 2x + 3. When it is divided by Additional Example 6.13 The polynomial is divisible by 2x + 3. When it is divided by x  1, the remainder is 15. Find the values of p and q. Solution: Since g(x) is divisible by 2x + 3, the remainder is 0.

When g(x) is divided by x - 1, the remainder is -15. Additional Example 6.13 When g(x) is divided by x - 1, the remainder is -15. Substitute p = 3 into (2), Solution:

Additional Example

Additional Example

Additional Example

Let Q(x) be the quotient and cx + d be the remainder when Additional Example 6.14 Without doing an actual division, find the remainder when is divided by Solution: When x = 5, Let Q(x) be the quotient and cx + d be the remainder when is divided by (x - 5)(x + 3). When x = -3,

Additional Example 6.14 Substitute c = 35 into (1), Solution:

Additional Example

Additional Example

Factorize the following polynomials. Additional Example 6.15 Factorize the following polynomials. Solution: [ The coefficient of x3 is 1 and the factors of the constant term 18 are 1, 2, 3, 6, 9, 18.  The possible linear factors of f (x) are x  1, x  2, x  3, x  6, x  9, x  18.] By the factor theorem,  x - 1 is not a factor of f (x).  x + 1 is not a factor of f (x).  x - 2 is a factor of f (x).

Solution: By long division, Additional Example 6.15 By long division, Solution: [ The factors of the coefficient of x3 are 1, 2, 3, 6 and the constant term is 1.  The possible linear factors of f (x) are x  1, 2x  1, 3x  1, 6x  1.]

Solution: By the factor theorem,  x - 1 is a factor of f (x). Additional Example 6.15 By long division, Solution: By the factor theorem,  x - 1 is a factor of f (x).

Additional Example

Additional Example

Factorize the following polynomials. Additional Example 6.16 Factorize the following polynomials. Solution: [ The factors of the coefficient of x3 are 1, 3 and the factors of the constant term -8 are 1, 2, 4, 8.  The possible linear factors of f (x) are x  1, x  2, x  4, x  8, 3x  1, 3x  2, 3x  4, 3x  8.] By the factor theorem,  x - 1 is not a factor of f (x).  x + 1 is not a factor of f (x).

 x - 2 is not a factor of f (x). Additional Example 6.16  x - 2 is not a factor of f (x).  x + 2 is a factor of f (x). By long division, Solution:

 x - 1 is not a factor of f (x). Additional Example 6.16  The factors of the coefficient of x3 are 1, 2 and the factors of the constant term are 1, 2, 4, 8, 16.  The possible linear factors of f (x) are x  1, x  2, x  4, x  8 , x  16 , 2x  1. By the factor theorem,  x - 1 is not a factor of f (x).  x + 1 is not a factor of f (x).  x - 2 is not a factor of f (x).  x + 2 is not a factor of f (x).  x - 4 is a factor of f (x). Solution:

Additional Example 6.16 By long division, Solution:

Additional Example

Additional Example

Additional Example 6.17 Determine whether each of the following polynomials has a linear factor with integral coefficient and constant term. Solution: (a) [ The factors of the coefficient of x3 are 1, 2 and the factors of the constant term are 1, 3, 5, 15.  The possible linear factors with integral coefficient and constant term of the polynomial are x  1, x  3, x  5, x  15, 2x  1, 2x  3, 2x  5, 2x  15.] By the factor theorem,

Additional Example 6.17 Solution:

Additional Example 6.17 Solution:

Additional Example 6.17 (b)  The factors of the coefficient of x3 are 1, 3 and the constant term is 1.  The possible linear factors with integral coefficient and constant term of the polynomial are x  1, 3x  1. By the factor theorem, Solution: