Grade 7: Algebraic Expressions

2. :
In this lesson, students should be able to:
1) multiply polynomials such as;
a) monomial by monomial,
b) monomial by polynomial with more than one term,
c) binomial by binomial,
d) polynomial with more than one term to polynomial with three or more terms.
2) solve problems involving multiplying polynomials. 2

3. A. To multiply a monomial by another monomial, simply
multiply the numerical coefficients then multiply the literal
coefficients by applying the basic laws of exponent.
( 3•-5 )
Examples:
1) (x3)(x5) = x8
2) (3x2)(-5x10) =
3) (-8x2y3)(-9xy8) =
= -15x12
= 72x3y11
( x2•x10)
( -8•-9 )( x2•x)( y3•y8)
Rules in Multiplying Polynomials

4. B. To multiply monomial by a polynomial, simply apply the
distributive property and follow the rule in multiplying
monomial by a monomial.
=(3x•x2)
Examples:
1) 3x (x2 – 5x + 7)
2) -5x2y3 (2x2y–3x+4y5)=
= 3x3 - 15x2 + 21x
(-5x2y3•2x2y)
Rules in Multiplying Polynomials
(3x•-5x) (3x•7)
(-5x2y3•-3x)(-5x2y3•4y5)
= -10x4y4 + 15x3y3 - 20x2y8

5. C.To multiply binomial by another binomial, simply distribute the first term of the
first binomial to each term of the other binomial then distribute the second term
to each term of the other binomial and simplify the results by combining similar
terms.
First =(x•x) = x2
Example:
Outer =(x•5)= 5x
x2+8x+15
FOIL METHOD OR SMILE METHOD
First
Outer
(x2+5x+3x+15)
1) ( x + 3 ) ( x + 5 )
Inner
Last
Inner =(3•x)= 3x
Last =(3•5) =15

6. D.To multiply a polynomial with more than one term by a polynomial with
three or more terms, simply distribute the first term of the first
polynomial to each term of the other polynomial. Repeat the procedure up
to the last term and simplify the results by combining similar terms.
x(x2 – 2x + 3) – 3(x2 – 2x + 3)
Example:
1) (x – 3)(x2 – 2x + 3) =
Rules in Multiplying Polynomials
= x3 – 2x2 + 3x – 3x2 + 6x – 9
= x3 – 5x2 + 9x – 9

7. = x2(4x3 + 5x – 1) + 3x(4x3 + 5x – 1) - 4(4x3 + 5x – 1)
Example:
2) (x2 + 3x – 4)(4x3 + 5x – 1)
Rules in Multiplying Polynomials
= 4x5 + 5x3 – x2 + 12x4 + 15x2 – 3x – 16x3 – 20x + 4
= 4x5 + 12x4 – 11x3 + 14x2 – 23x + 4

8. = (6x2 – 5x – 6)(x2 – 2x – 1)
Example:
3) (2x – 3)(3x + 2)(x2 – 2x – 1)
= 6x4 – 17x3 – 2x2 + 17x + 6
= 2x(3x + 2) – 3(3x + 2) (x2 – 2x – 1)
= (6x2 + 4x – 9x – 6) (x2 – 2x – 1)
= 6x2(x2 – 2x – 1) – 5x(x2 – 2x – 1)– 6(x2 – 2x – 1)
= 6x4 – 12x3 – 6x2 – 5x3 + 10x2 + 5x – 6x2 +12x + 6

9. = (2x)3 + 3(2x)2(5) + 3(2x)(5)2 + (5)3
Examples: Cube of a Binomial
A} (x + y)3 = x3 + 3x2y + 3xy2 + y3
B} (x – y)3 = x3 – 3x2y + 3xy2 – y3
Example # 1 : (2x + 5)3
= 8x3 + 60x2 + 150x + 125
Example # 2 : (3x – 2y)3
= (3x)3 – 3(3x)2(2y) + 3(2x)(2y)2 – (2y)3
= 27x3 – 54x2y + 24xy2 – 8y3

10. = (x)2 + (2y)2 + (5z)2 + 2(x)(2y) + 2(x)(5z) + 2(2y)(5z)
Examples: Square of a Trinomial
A} (a + b + c)2 = a2 + b2 +c2 + 2ab +2ac + 2bc
Example # 1 : (x + 2y + 5z)2
= (x)2 + 4y2 + 25z2 + 4xy + 10xz + 20yz
Example # 2 : (2x – 3y + z)2
= (2x)2 + (–3y)2 + (z)2 + 2(2x)(–3y) + 2(2x)(z) + 2(–3y)(z)
= 4x2 + 9y2 + z2 –12xy + 4xz – 6yz

11. Find each indicated product.
a. 3(x
2
+ 7)
b. 2s( s - 4 )
c. (w + 9) ( w - 2)
d. (4f + 1 ) ( f - 5)
e. (x - 4 ) ( x + 4)
f. (2x - 3)
2
= 3x2 + 21
= 2s2 − 8s
= w2 + 7w - 18
= 4f2 − 19f - 5
= x2 − 16
= 4x2 − 12x + 9

12. Find each indicated product.
a. 2(x
2
− 3)
b. 3s( s + 5 )
c. (w + 4) ( w - 1)
d. (2n + 1 ) ( n - 5)
e. (x - 7) ( x + 7)
f. (3x - 2)
2
= 2x2 − 6
= 3s2 + 15s
= w2 + 3w - 4
= 2n2 − 9n - 5
= x2 − 49
= 9x2 − 12x + 4

14. Example (6x2 – 5x – 6)(x2 – 2x – 1)
Step 1 : Arrange the polynomial in a descending order.
6 – 5 – 6 x 1 – 2 – 1
Step 2 : Write down the numerical coefficient of each polynomial.
note: supply 0 if there is a missing term in the degree of polynomial.
6 -5 -6
1
-2
-1
6 -5 -6
-12 10 12
-6 5 6
6
-17
-2
17 6
Step 7 : Add the exponents of the first term of each
polynomial, then start arranging it in
descending order.
6 – 17 – 2 + 17 + 6
Step 3 : Prepare a box patteren as required. Box Formula = Highest degree +1 ; since (3 X 3) +1 = 4 x 4
Step 4 : Place the first numerical values in the upper row and the second numerical values in the last column.
Step 6 : Add the numbers diagonally.
6x4 – 17x3 – 2x2 + 17x + 6
Step 5 : Multiply the column by its row and place the product
always on top of each corresponding square.

15. Example: (– 2x3 – 4x + 1) (x2 – 3x – 2)
-2 0 -4 1
1
-3
-2 0 -4 1
6 0 12 -3
4
-2
6
0
13 5
– 2 +6 +0 +13 +5 – 2
– 2x3 + 0x2 – 4x + 1) (x2 – 3x – 2)
(– 2 + 0 – 4 + 1) (1 – 3 – 2)
-20 8 -2
-2
– 2x5 + 6x4 + 0x3 + 13x2 + 5x – 2
– 2x5 + 6x4 + 13x2 + 5x – 2