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:

8wMyDImAAA7

  • 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

K1icOtq0shWo2C6tXCyWSsUr7VFM7aDS + HqR4P + IPXeZaQm1qS3o6e6tbJJjPAPrxPk68qNf + NNnkdGVuNr3M9x6lp7eXem9wteIMzvB40JoJ6aEorNtsP0lKT1rVgZOrFbq + lylFoc62tIlLuNYIxoZZxphAfBXFhJRJmCrqox2bSx2f + VvtdpV54lJiVY93WPFQzD46Gl9eIcK84rgSQdsVT109S6xzCearCZmUKfNlV +7 aLXPbPJbwunmZxrU2Us2blJMwVBfaHe3B0bIWZhUrzcsK6F8RzGJAAAA7

  • y4 · y6 = y4 + 6 = y10

nWYSVcwQmRh2DciFYnQEFAAA7

(-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

kKJmXvBT + vwFlzKKGu0GyNEAAAAASUVORK5CYII =

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:

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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.

v4BXMEukYaChA5W2TcFl7WJDpHgYFEAADs =

AALDhI + pF +0 Pw5q0WvaEiKP7Hl3i6GjmqT3f6jXjqzTovEkYy7owGdCoHgvgVsDdpOc72SrC4WdpTCCTKZLzGZVSTdBL8zro7qZUsRcMypKTZtH32mZuq1nA2xlfrH15K7qPsEcDePZHMrdRJ4UW5jZH6AcGKDgDGQlnQfmjGMOYp6nEqedJAcolGnE3lGpjmoiqykgk4foqaie7yoBo6RYrCVXby5MrJnyr94sV84hcitfmOhzVkFPK65zZVBTECpHtBVgAADs =

h1uwYxoUJJPIwUAAA7

Examples:

  • AAKajH + gy932hJwiuOvCsCzQyWGYNmzdJ4ViVpqLh6rr25YqTMkzWbsADpp1eq0QMCVU8IiuYyX5Y9YsTt1lKaXGkggi4od6QrE +8 JbLlFXH6cjHevVmzkKyzineyefvepuf49cDhwdHMyiylrg3QheHuKLIwtjY9zjF5aj0txOmRtlpuQkJZRYYRVpqCmKn + mCQY7AB4XqFkABVAAA7
  • AAKujI + pB +0 Po5y0ymVjELz7DwpBRpamkwzqOiDlFsbfeNY2gLD63hoWLAuKbsRJiofkuS7CIK1IzCWn1J7vAWyCntDXQlpVfhkNgzbG7RYDYWQ6kz133uoTu72jw + + H3h3UnuJJoYngoQqi42OhIqXYgyZFZF3jZU6kRyUe2dhB6OnqxebjkxSgIy0qS8EqLc4SXUKs5FtcHnDfma0wKnKxwXFEAADs = VbfZ1igKdzdWPG5
  • e7 + fa1I6X5ZILLVRvu175Ca6ytwxtPTispEM3FfhkEJbl9zJ4VLh2AOhn5GOyxnWYFjjZU2m5o5AXFaPgWVp16uhZAAA7
  • (2y + 5) (y2 + 4y – 7) = 2y3 + 8y2 – 14y + 5y2 + 20y – 35 = 2y3 + (8y2 + 5y2) + (-14y + 20y) – 35 = 2y3 + 13y2 + 6y – 35

Ot07nVXDbUW3dwDgUAADs =

  • AAKajH + gy932hJwiuOvCsCzQyWGYNmzdJ4ViVpqLh6rr25YqTMkzWbsADpp1eq0QMCVU8IiuYyX5Y9YsTt1lKaXGkggi4od6QrE +8 JbLlFXH6cjHevVmzkKyzineyefvepuf49cDhwdHMyiylrg3QheHuKLIwtjY9zjF5aj0txOmRtlpuQkJZRYYRVpqCmKn + mCQY7AB4XqFkABVAAA7
  • AAKujI + pB +0 Po5y0ymVjELz7DwpBRpamkwzqOiDlFsbfeNY2gLD63hoWLAuKbsRJiofkuS7CIK1IzCWn1J7vAWyCntDXQlpVfhkNgzbG7RYDYWQ6kz133uoTu72jw + + H3h3UnuJJoYngoQqi42OhIqXYgyZFZF3jZU6kRyUe2dhB6OnqxebjkxSgIy0qS8EqLc4SXUKs5FtcHnDfma0wKnKxwXFEAADs = VbfZ1igKdzdWPG5
  • e7 + fa1I6X5ZILLVRvu175Ca6ytwxtPTispEM3FfhkEJbl9zJ4VLh2AOhn5GOyxnWYFjjZU2m5o5AXFaPgWVp16uhZAAA7
  • (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