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Multiplying
Polynomials
I will multiply polynomial expressions using the
distributive property and exponent rules
Multiply.
Example 1: Multiplying Monomials
A. (6y3)(3y5)
(6y3)(3y5)
18y8
Group factors with like bases
together.
B. (3mn2) (9m2n)
(3mn2)(9m2n)
27m3n3
Multiply.
Group factors with like bases
together.
Multiply.
(6 3)(y3 y5)
(3 9)(m m2)(n2 n)
Multiply.
Example 1C: Multiplying Monomials
2 2 2112
4s t st st 4 53s t
Group factors with like
bases together.
Multiply.
22 2112
4ts tt s s
g gg g g
2 2112
4ts ts ts
2
When multiplying powers with the same base, keep the base and add the exponents.
x2 x3 = x2+3 = x5
Remember!
Check It Out! Example 1
Multiply.
a. (3x3)(6x2)
(3x3)(6x2)
(3 6)(x3 x2)
18x5
Group factors with like bases
together.
Multiply.
Group factors with like bases
together.
Multiply.
b. (2r2t)(5t3)
(2r2t)(5t3)
(2 5)(r2)(t3 t)
10r2t4
Check It Out! Example 1
Multiply.
Group factors with
like bases
together.
Multiply.
c.
4 52 2112
3x zy zx y
3
2112
3x y x z y z
2 4 53
g gg g
3 22 4 5112 z
3zx x y y
7554x y z
To multiply a polynomial by a monomial, use the Distributive Property.
6pq(2p – q)
(6pq)(2p – q)
Multiply.
Example 2B: Multiplying a Polynomial by a Monomial
(6pq)2p + (6pq)(–q)
(6 2)(p p)(q) + (–1)(6)(p)(q q)
12p2q – 6pq2
Distribute 6pq.
Group like bases
together.
Multiply.
Check It Out! Example 2
Multiply.
a. 2(4x2 + x + 3)
2(4x2 + x + 3)
2(4x2) + 2(x) + 2(3)
8x2 + 2x + 6
Distribute 2.
Multiply.
Check It Out! Example 2
Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
(3ab)(5a2) + (3ab)(b)
(3 5)(a a2)(b) + (3)(a)(b b)
15a3b + 3ab2
Distribute 3ab.
Group like bases
together.
Multiply.
Check It Out! Example 2
Multiply.
c. 5r2s2(r – 3s)
5r2s2(r – 3s)
(5r2s2)(r) – (5r2s2)(3s)
(5)(r2 r)(s2) – (5 3)(r2)(s2 s)
5r3s2 – 15r2s3
Distribute 5r2s2.
Group like bases
together.
Multiply.
To multiply a binomial by a binomial, you can apply the Distributive Property more than once:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3.
Distribute x and 3
again.
Multiply.
Combine like terms.
= x(x + 2) + 3(x + 2)
= x(x) + x(2) + 3(x) + 3(2)
= x2 + 2x + 3x + 6
= x2 + 5x + 6
Multiply.
Example 3A: Multiplying Binomials
(s + 4)(s – 2)
(s + 4)(s – 2)
s(s – 2) + 4(s – 2)
s(s) + s(–2) + 4(s) + 4(–2)
s2 – 2s + 4s – 8
s2 + 2s – 8
Distribute s and 4.
Distribute s and 4
again.Multiply.
Combine like terms.
Multiply.
Example 3B: Multiplying Binomials
(x – 4)2
(x – 4)(x – 4)
(x x) + (x (–4)) + (–4 x) + (–4 (–4))
x2 – 4x – 4x + 8
x2 – 8x + 8
Write as a product of
two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
Example 3C: Multiplying Binomials
Multiply.
(8m2 – n)(m2 – 3n)
8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)
8m4 – 24m2n – m2n + 3n2
8m4 – 25m2n + 3n2
Use the FOIL method.
Multiply.
Combine like terms.
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint
Check It Out! Example 3a
Multiply.
(a + 3)(a – 4)
(a + 3)(a – 4)
a(a – 4)+3(a – 4)
a(a) + a(–4) + 3(a) + 3(–4)
a2 – a – 12
a2 – 4a + 3a – 12
Distribute a and 3.
Distribute a and 3
again.
Multiply.
Combine like terms.
Check It Out! Example 3b
Multiply.
(x – 3)2
(x – 3)(x – 3)
(x x) + (x(–3)) + (–3 x)+ (–3)(–3)●
x2 – 3x – 3x + 9
x2 – 6x + 9
Write as a product of
two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
Check It Out! Example 3c
Multiply.
(2a – b2)(a + 4b2)
(2a – b2)(a + 4b2)
2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)
2a2 + 8ab2 – ab2 – 4b4
2a2 + 7ab2 – 4b4
Use the FOIL method.
Multiply.
Combine like terms.
To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):
(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)
= 10x3 + 50x2 – 30x + 6x2 + 30x – 18
= 10x3 + 56x2 – 18
Multiply.
Example 4A: Multiplying Polynomials
(x – 5)(x2 + 4x – 6)
(x – 5 )(x2 + 4x – 6)
x(x2 + 4x – 6) – 5(x2 + 4x – 6)
x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)
x3 + 4x2 – 5x2 – 6x – 20x + 30
x3 – x2 – 26x + 30
Distribute x and –5.
Distribute x and −5
again.
Simplify.
Combine like terms.
Check It Out! Example 4a
Multiply.
(x + 3)(x2 – 4x + 6)
(x + 3 )(x2 – 4x + 6)
x(x2 – 4x + 6) + 3(x2 – 4x + 6)
Distribute x and 3.
Distribute x and 3
again.
x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)
x3 – 4x2 + 3x2 +6x – 12x + 18
x3 – x2 – 6x + 18
Simplify.
Combine like terms.
Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
(3x + 2)(x2 – 2x + 5)
x2 – 2x + 53x + 2
Multiply each term in the
top polynomial by 2.
Multiply each term in the
top polynomial by 3x,
and align like terms.2x2 – 4x + 10+ 3x3 – 6x2 + 15x
3x3 – 4x2 + 11x + 10Combine like terms by
adding vertically.
Lesson Quiz: Part I
Multiply.
1. (6s2t2)(3st)
2. 4xy2(x + y)
3. (x + 2)(x – 8)
4. (2x – 7)(x2 + 3x – 4)
5. 6mn(m2 + 10mn – 2)
6. (2x – 5y)(3x + y)
4x2y2 + 4xy3
18s3t3
x2 – 6x – 16
2x3 – x2 – 29x + 28
6m3n + 60m2n2 – 12mn
6x2 – 13xy – 5y2
Lesson Quiz: Part II
7. A triangle has a base that is 4cm longer than its height.
a. Write a polynomial that represents the area of the triangle.
b. Find the area when the height is 8 cm.
48 cm2
1
2h2 + 2h