1.1Index Notation 1.2Concepts of Polynomials 1Manipulations and Factorization of Polynomials 1.3Addition and Subtraction of Polynomials 1.4Multiplication of Polynomials 1.5Factorization by Taking Out Common Factors and Grouping Terms

1.1Index Notation A.Introduction to Indices We can compare it with a = a 1, which is the 1st power of a. a 2 can be read as ‘a squared’. a 3 can be read as ‘a cubed’. a = a 1, which is the 1st power of a. a 2 can be read as ‘a squared’. a 3 can be read as ‘a cubed’.

1.1Index Notation A.Introduction to Indices

Solution: Represent each of the following expressions using index notation. (a)6  6  6  7  7 (b)h  h  k  k  k (a)6  6  6  7  7 (b) h  h  k  k  k Example 1T 1Manipulations and Factorization of Polynomials

Solution: Use index notation to represent each of the following expressions. (a)(  5)  (  d)  (  5)  (  d)  (  d)  (  5)  d  (  5) (b)m  (  m)  n  m  (  n)  (  m)  (  m) (a) (  5)  (  d)  (  5)  (  d)  (  d)  (  5)  d  (  5)   (5  d  5  d  d  5  d  5)   (5  5  5  5  d  d  d  d) (b) m  (  m)  n  m  (  n)  (  m)  (  m)  m  m  n  m  n  m  m  m  m  m  m  m  n  n Example 2T 1Manipulations and Factorization of Polynomials

1.1Index Notation B.Addition of Indices The base of an expression remains unchanged during simplification, whether it is a variable or a number.

Solution: Simplify each of the following expressions. (a)5d 2  2d 8 (b) (5p 5 )(3p) (c)  u 2  4u 3  9u 4 (a) (c) (b)(b) Example 3T 1Manipulations and Factorization of Polynomials

Solution: Simplify each of the following expressions. (a)(  6p 5 q 3 )(4p 2 )(b)(7h 2 k 4 )(8k 5 h 3 ) (b) (a) Example 4T 1Manipulations and Factorization of Polynomials

1.1Index Notation C.Subtraction of Indices If a = 0, then a n = 0. In this case, a m  a n is undefined because a m is divided by 0.

Simplify the following expressions. (a)(b) Solution: (a) (b)(b) Example 5T 1Manipulations and Factorization of Polynomials

Solution: Simplify 6t 6  18t 3  9t 4. Example 6T 1Manipulations and Factorization of Polynomials

1.2Concepts of Polynomials A.Definition

1.2Concepts of Polynomials A.Definition

1.2Concepts of Polynomials B.Degree and Arrangement of Terms

1.2Concepts of Polynomials B.Degree and Arrangement of Terms

Determine the degree of each of the following polynomials. (a) 6a 3  5a 5 + 1(b)  5h 2 k 3 + 6l  8m 6 Solution: (a) ∵ Degrees of monomials 6a 3, 5a 5 and 1 are 3, 5 and 0 respectively. ∴ Degree of 6a 3  5a 5  1 (b) ∵ Degrees of monomials  5h 2 k 3, 6l and 8m 6 are 5, 1 and 6 respectively. ∴ Degree of  5h 2 k 3 + 6l  8m 6 Example 7T 1Manipulations and Factorization of Polynomials

Arrange the terms of each of the following polynomials in descending powers of a. (a)  2b  a 3  3ab(b)ab 2 c  5a 2 c  b 4 (a)  2b  a 3  3ab  a 3  3ab  2b Solution: (b)ab 2 c  5a 2 c  b 4   5a 2 c  ab 2 c  b 4  a 3  3ba  2b   5ca 2  b 2 ca  b 4 Example 8T 1Manipulations and Factorization of Polynomials

1.2Concepts of Polynomials C.Values of Polynomials

Find the value of the polynomial 3x  5y  x 2 y in each of the following cases. (a)x  1, y   5(b)x  4, y  3 Solution: Example 9T 1Manipulations and Factorization of Polynomials

1.3Addition and Subtraction of Polynomials A.Like Terms and Unlike Terms The coefficients of the terms are not involved in classifying like terms.

First remove the brackets, then combine the like terms. 1.3Addition and Subtraction of Polynomials B.Addition and Subtraction of Polynomials

Simplify the following polynomials. (a)(7c  2d)  (  3c  2d)(b)(8y  z)  (  z  y) Solution: (a) (b) Example 10T 1Manipulations and Factorization of Polynomials

Simplify the following polynomials. (a)(  9r 3  6r 2  9r  8)  (5  10r  3r 2  8r 3 ) (b)(2x 3  x 2  9x)  (  4x 4  7x 3  3x 2 ) Solution: (a) (b) Example 11T 1Manipulations and Factorization of Polynomials

Simplify the following polynomials. (a)(3a 2 b  4ab  5a)  (2a 2 b  7a 2  2ab  5) (b)(7r 2  4rs  5s)  (  8s  12  6r 2 ) Solution: Example 12T 1Manipulations and Factorization of Polynomials

1.4Multiplication of Polynomials A.Multiplication of Polynomials

1.4Multiplication of Polynomials A.Multiplication of Polynomials

(b) (4r  s)(6r)  (4r)(6r)  (s)(6r) Solution: (a) Expand the following polynomials. (a)  2(  h  5k) (b) (4r  s)(6r) (c) (2z 2 + z – 3)(8) (c)(c) Example 13T 1Manipulations and Factorization of Polynomials

Expand the following polynomials. (a) (u  3)(4u  5) (b)(  z  2)(z  3) Solution: Example 14T 1Manipulations and Factorization of Polynomials

Expand the following polynomials. (a)(xy  1)(4xy  3)(b)(5ab  2)(2a  b  4) Solution: Example 15T 1Manipulations and Factorization of Polynomials

1.4Multiplication of Polynomials B.Mixed Operations of Polynomials

Simplify 2y(y – 3) + 3(y 2 – 1). Solution: Example 16T 1Manipulations and Factorization of Polynomials

Simplify [(x  3y)(2x  5y)  2](  x). Solution: Example 17T 1Manipulations and Factorization of Polynomials

1.5Factorization by Taking Out Common Factors and Grouping Terms

A.Taking Out Common Factors

Factorize the following expressions. (a) 6xy + 9xz (b) 2y 2 + 3y 3 (c) 15m 2 n + 12n (d) – p – p 2 Solution: (a) 6xy  9xz  3x(2y)  3x(3z) (b)(b) (c)(c) (d)(d) Example 18T 1Manipulations and Factorization of Polynomials

Factorize the following expressions. (a) 5a 2 x + 15a 2 x 2 (b) 45pq – 60pqr (c) 18x 3 y + 24x 2 y 2  30xy 3 Solution: (a) (b) (c) Example 19T 1Manipulations and Factorization of Polynomials

Factorize the following expressions. (a) (2a – b)c + (2a – b)d(b) 2m(x – 2y) + 4n(2y – x) (c) –5rt – 2t(3r – 4s)(d) 18m 2 n(p – q) 2 – 27mn 2 (q – p) Solution: (a) (b) (c) (d) Example 20T 1Manipulations and Factorization of Polynomials

1.5Factorization by Taking Out Common Factors and Grouping Terms B.Grouping Terms

Factorize the following expressions. (a) 2ab + 6b + ac + 3c (b) 6h – 2hm + 3k – km Solution: Example 21T (a) 2ab  6b  ac  3c  (2ab  6b)  (ac  3c)  2b(a  3)  c(a  3) (b) 6h  2hm  3k  km  (6h  3k)  (2hm  km)  3(2h  k)  m(2h  k) 1Manipulations and Factorization of Polynomials

Factorize the following expressions. (a) 1 – y + 5xy – 5x (b) 6xyz – 6wyz – 5w + 5x Solution: Example 22T (a) (b)(b) 1Manipulations and Factorization of Polynomials

Factorize the following expressions. (a) x 2 y 2 + y 2 + x (b) 3az – 6bz – 12b + 6a Solution: Example 23T (a) (b)(b) 1Manipulations and Factorization of Polynomials

Follow-up 1 Represent each of the following expressions using index notation. (a)3  3  3  4  4  4 (b)p  p  q  q  q  q Solution: (a)3  3  3  4  4  4 (b)p  p  q  q  q  q 1Manipulations and Factorization of Polynomials

Follow-up 2 Use index notation to represent each of the following expressions. (a)(–7)  (–7)  (–a)  (–7)  (–a)  a  ( –7) (b)c  (–d)  c  d  (–d)  (–c)  (–c)  d  (–d) Solution:  7  7  a  7  a  a  7  7  7  7  7  a  a  a (a) 1Manipulations and Factorization of Polynomials (b)(b) c  (  d)  c  d  (  d)  (  c)  (  c)  d  (  d)   (c  d  c  d  d  c  c  d  d)   (c  c  c  c  d  d  d  d  d)

Follow-up 3 Simplify the following expressions. (a)4c 3  3c 4 (b)2n 5  5n 2 (c)(  2z)(4z 4 )(6z 7 ) Solution: (a) (b) 1Manipulations and Factorization of Polynomials (c)

Follow-up 4 Solution: Simplify the following expressions. (a) 18u 7 v 4  9v 5 u(b) (  4ab 2 )(  3a 3 b 6 ) (a) 18u 7 v 4  9v 5 u (b) (  4ab 2 )(  3a 3 b 6 ) 1Manipulations and Factorization of Polynomials

Follow-up 5 Simplify the following expressions. (a) (b)8y 8  y 4  2y 2 Solution: (a) (b) 1Manipulations and Factorization of Polynomials

Follow-up 6 Solution: (a)(b) Simplify the following expressions. (a)(b)  4r 5  12r 3  3r 2 1Manipulations and Factorization of Polynomials

Follow-up 7 Determine the degree of each of the following polynomials: Solution: Degrees of monomials of  5m and 6m 2 are 1 and 2 respectively. Degree of 3  5m + 6m 2 (a) (b) Degrees of monomials of 3p 3,  5p 2 q 2 and  8q 2 are 3, 4 and 2 respectively. Degree of 3p 3  5p 2 q 2  8q 2 1Manipulations and Factorization of Polynomials

Follow-up 8 Solution: Arrange the terms of each of the following polynomials in descending powers of s. (a)r  6sr  5r 2 s 2 (b)  3  2ts 2  st 3 w (a)r  6sr  5r 2 s 2   5r 2 s 2  6sr  r (b)  3  2ts 2  st 3 w   2ts 2  st 3 w  3 1Manipulations and Factorization of Polynomials

Follow-up 9 Solution: Find the values of the following polynomials when h  1 and k   2. (a)9k  hk  4k 2 h(b)  3  2h 3  6kh 2  k 1Manipulations and Factorization of Polynomials

Follow-up 10 Solution: Simplify the following polynomials. (a)(5m  7n)  (  8n  5m)(b)(  4v  10u)  (4u  9v) (a) (b) 1Manipulations and Factorization of Polynomials

Follow-up 11 Solution: Simplify the following polynomials. (a)(5y 3  4  2y)  (2y 2  4y 3  1  2y) (b)(3a 2  a  3)  (3a  a 2  3) 1Manipulations and Factorization of Polynomials

Follow-up 12 Solution: Simplify the following polynomials. (a)(5yz  4y 2 z  z  10)  (2y 2 z  6  6z) (b)(4mn 2  5m 2  8)  (  2mn 2  3n 2  4) 1Manipulations and Factorization of Polynomials

Follow-up 13 Expand the following polynomials. (a)4(  a  4b)(b)(3m  4n)  2n (c) u(  2 + 4h  k) (a)4(  a  4b)  4(  a)  4(4b) (b)(3m  4n)  2n  (3m)(2n)  (4n)(2n) Solution: (c)u(  2  4h  k)  u(  2)  u(4h)  uk 1Manipulations and Factorization of Polynomials

Follow-up 14 Expand the following polynomials. (a)(2x  3)(x  4) (b)(  5y  1)(5y  1) Solution: 1Manipulations and Factorization of Polynomials

Follow-up 15 Solution: Expand the following polynomials. (a)(4m  n)(2n  3m)(b)(4y  1)(5y 2  5y  4) 1Manipulations and Factorization of Polynomials

Follow-up 16 Solution: Simplify 6m – 5m(m – 4). 6m  5m(m  4)  6m  5m(m)  5m(  4)  6m  5m 2  20m   5m 2  6m  20m 1Manipulations and Factorization of Polynomials

Follow-up 17 Solution: Simplify the following polynomials. (a)(3a 2  4ab)  (2a  b)(a  4b) (b)(  2  5k)[(4k  3h)  (6h  2k)] 1Manipulations and Factorization of Polynomials

Follow-up 17 Solution: Simplify the following polynomials. (a)(3a 2  4ab)  (2a  b)(a  4b) (b)(  2  5k)[(4k  3h)  (6h  2k)]

Follow-up 18 Solution: Factorize the following expressions. (a)4mn – 12mp (b) x 5 + 4x (c)9p 4 – 3p 2 (d) –2x – 4y (a) 4mn  12mp  4m(n)  4m(3p) (b) x 5 + 4x  x(x 4 ) + x(4) (c)(c) (d)(d)  2x  4y   2(x)  (  2)(2y) 1Manipulations and Factorization of Polynomials

Follow-up 19 Solution: Factorize the following expressions. (a)3a 2 x – 12ax 2 (b) 25a 2 c + 40abc 2 (c)14m 3 n – 28m 2 n m 2 n 3 1Manipulations and Factorization of Polynomials

Follow-up 20 Solution: Factorize the following expressions. (a)(a – c)b + (a – c)d (b) 5m(p – q) – n(q – p) (c)(4x – 3y)z + 5yz 1Manipulations and Factorization of Polynomials

Follow-up 21 Solution: Factorize the following expressions. (a)3xz – x + 6yz – 2y (b) 4mp – 4np + mq – nq (c)6ac – 4a – 9bc + 6b (a) 3xz  x  6yz  2y  (3xz  6yz)  (x  2y)  3z(x  2y)  (1)(x  2y) (b) 4mp  4np  mq  nq  (4mp  4np)  (mq  nq)  4p(m  n)  q(m  n) 1Manipulations and Factorization of Polynomials (c)6ac  4a  9bc  6b  (6ac  9bc)  (4a  6b)  3c(2a  3b)  2(2a  3b)

Follow-up 22 1Manipulations and Factorization of Polynomials Solution: Factorize the following expressions. (a)6ac + 6ac + a + b (b) 4rst – 6rsu + 2t – 3u (a) 6ac  6bc  a  b  (6ac  6bc)  (a  b)  6c(a  b)  (1)(a  b) (b)(b)

Follow-up 23 Solution: Factorize the following expressions. (a)2b + ab – a – 2b 2 (b) –2x 2 y + 6xz + 12xy – 36z (a)(a) (b)(b) 1Manipulations and Factorization of Polynomials