Mastering Polynomial Multiplication: Rules and Examples
Rules of Exponents for Multiplication of Polynomials
Rule #1: Rule of Exponent 1
If a is a real number, then a = a1.
This rule states that when a variable has no exponent, its exponent is one (1).
Examples:
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- 51 = 5
- (-6)1 = -6
- (3a3b4)1 = 3a3b4
Rule #2: Multiplication of Powers
If a is any real number and m, n are positive integers, then
am · an = am + n
This means that when we multiply exponents with equal bases, we add the exponents.
Examples:
- x7 · x8 = x7 + 8 = x15
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- y4 · y6 = y4 + 6 = y10
(-4)2 · (-4) = (-4)2 + 1 = (-4)3
Rule #3: Power of a Power
If a is a real number and m, n are positive integers, then
(an)m = an * m
Examples:
- (a3)5 = a3 · 5 = a15
- (p4)6 = p4 · 6 = p24
Rule #4: Power of a Product
If a, b are real numbers and n is a positive integer, then
(ab)n = anbn
Examples:
- (ab)4 = a(1 • 4) b(1 • 4) = a4 b4
- (3m)4 = 3(1 • 4) m(1 • 4) = 34 m4 = 81m4
- (-2p3q5)2 = (-2)2 p(3 • 2) q(5 • 2) = 4 p6 q10
Multiplication of a Monomial by Another Monomial
To multiply monomials, we find the product of the coefficients. Then apply the rules of exponents for multiplication of equal bases.
Remember the rules of signs in multiplication and division:
(Positive) (Positive) = Positive (Positive Signs are equal)
(Negative) (Negative) = Positive
(Positive) (Negative) = Negative (Difference is negative signs)
(Negative) (Positive) = Negative
Examples:
- (5x5y2) (-4x3y7) = (5) (-4) x(5 + 3) y(2 + 7) = -20x8y9
- (4m6n5t2) (-2m2n3t4) (3mn7t8) = (4) (-2) (3) m(6 + 2 + 1) n(5 + 3 + 7) t(2 + 4 + 8) = -24m9n15t14
- (12a6b2) (-4a5b4) (-b3) = (12) (-4) (-1) a(6 + 5) b(2 + 4 + 3) = 48a11b9
Practice:
Page 172, 1 to 50 odd
Monomial by Another Type of Polynomial
To multiply a monomial by another type of polynomial, use the distributive property and the laws of exponents.
Distributive Property: a (b + c) = ab + ac
Examples:
- -7 (2t + 5) = (-7) (2t) + (-7) (5) = -14t – 35
- y6 (7y2 + 2y – 6) = y6 (7y2) + y6 (2y) + y6 (-6) = 7y8 + 2y7 – 6y6
- 9m4 (-4m2 + 2m – 8) = 9m4 (-4m2) + 9m4 (2m) + 9m4 (-8) = -36m6 + 18m5 – 72m4
- 3x3y2z (-12x4y8z2 + 7x9z5) = 3x3y2z (-12x4y8z2) + 3x3y2z (7x9z5) = -36x7y10z3 + 21x12y2z6
Practice:
Page 179, 1 to 8
Binomial by Binomial
To multiply a binomial by another binomial, we use the distributive property.
Note: Remember to always add like terms.
Examples:
- (x + 3) (x + 7) = (x) (x) + (x) (7) + (3) (x) + (3) (7) = x2 + 7x + 3x + 21 = x2 + 10x + 21
- (3y + 7) (4y – 9) = (3y) (4y) + (3y) (-9) + (7) (4y) + (7) (-9) = 12y2 – 27y + 28y – 63 = 12y2 + y – 63
- (x + 5)2 = (x + 5) (x + 5) = (x) (x) + (x) (5) + (5) (x) + 25 = x2 + 5x + 5x + 25 = x2 + 10x + 25
Steps to get the square of a binomial:
- Square the first term.
- Multiply the terms of the binomial and then multiply by 2.
- Square the second term.
Examples:
- (x + y)2 = x2 + 2xy + y2
- (x – y)2 = x2 – 2xy + y2
Note: We can observe that the difference in these two examples is the sign of the middle term.
- (x – 6)2 = x2 – 12x + 36
- (4p + 3)2 = 16p2 + 24p + 9
Practice: Perform the following operations.
- (y + 4)2 =
- (2 – x)2 =
- (2p + 1)2 =
- (7x + 3y)2 =
- (-3y + 5)2 =
- (6a – 4)2 =
- (9a + b)2 =
- (6b – 7c)2 =
- (p – 10)2 =
- (5w + 7)2 =
Development of a Binomial Cubed
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Example:
- (x + 3)3 =
= (x)3 + 3 (x)2 (3) + 3 (x) (3)2 + (3)3
= x3 + 9x2 + 27x + 27
- (5t – 2)3 =
= (5t)3 + 3 (5t)2 (-2) + 3 (5t) (-2)2 + (-2)3
= 125t3 – 150t2 + 60t – 8
Practice: Develop the following pairs Cubed.
- (4 + w)3 =
- (7a + 2)3 =
- (h – g)3 =
- (8 + 3d)3 =
- (6y + x)3 =
- (4p – 9)3 =
- (5t – 4k)3 =
- (9 + 5g)3 =
- (8w – 3z)3 =
- (2n – 6m)3 =
Polynomial by Polynomial
To multiply a polynomial by another polynomial, we use the distributive property. That is, multiply each term in one of the polynomials by all the terms in the other polynomial and then add like terms.
Examples:
- (2y + 5) (y2 + 4y – 7) = 2y3 + 8y2 – 14y + 5y2 + 20y – 35 = 2y3 + (8y2 + 5y2) + (-14y + 20y) – 35 = 2y3 + 13y2 + 6y – 35
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- (7y – 6) (5y2 + 8y – 3) = 35y3 + 56y2 – 21y – 30y2 – 48y + 18 = 35y3 + (56y2 – 30y2) + (-21y – 48y) + 18 = 35y3 + 26y2 – 69y + 18
Practice:
Page 179, 53 to 64 odd