Lesson 3 · from §29, 30, 31, 32, 33, 34 of the 1910 text
Multiplying Polynomials
(5 + 3)(7 - 4) = 8 \cdot 3 = 24. We can also multiply piece by piece: 5(7 - 4) + 3(7 - 4) = 35 - 20 + 21 - 12 = 24. In the same way (2x + 3)(4x - 5) = 2x(4x - 5) + 3(4x - 5) = 8x^2 + 2x - 15, and in general (a + b)(c + d) = ac + ad + bc + bd.
Rule. Multiply the multiplicand by each term of the multiplier in turn, and add the partial products.
\begin{array}{rrrl} 3x & -\,2y & & \\ 2x & +\,3y & & \\ \hline 6x^2 & -\,4xy & & \text{multiplying by } 2x \\ & +\,9xy & -\,6y^2 & \text{multiplying by } 3y \\ \hline 6x^2 & +\,5xy & -\,6y^2 & \end{array}
Quick check
Multiply x + 5 by x - 2.
Powers, arrangement, and checking
A power of a number is the product of using it as a factor one or more times: 2^3 = 8 and (2x)^5 = 32x^5.
A polynomial is arranged in descending powers of x when the exponents of x get smaller from left to right, like 2x^4 - 5x^2 - 6x + 8; \;4 - 2y + y^2 is in ascending powers of y. Arrange both factors the same way before multiplying, and the partial products line up neatly in columns.
To check a product, give each letter a convenient value: the product of the factors' values should equal the product's value. Letting every letter be 1 is easiest, but it can't catch exponent mistakes (since 1^3 = 1^5). For a check on both coefficients and exponents, use 2.
Quick check
Check with x = 2: what is the value of x^2 + 3x - 10?
Degree and homogeneous expressions
The degree of a term in a letter is that letter's exponent: 3xy^2 is of the second degree in y. In several letters, add their exponents: 5x^3y is of the fourth degree in x and y. Terms are homogeneous if they have the same degree, like 3a^2b^3, 4ab^4, and a^4b. A polynomial is homogeneous if all its terms are.
The sum, difference, product, or quotient of two homogeneous expressions is homogeneous. That makes a quick exponent check: multiplying x^2 - 2xy + y^2 by x^3 - 3x^2y + 3xy^2 - y^3, every term of the answer must be of the fifth degree.
Typing tip: type fractions like 1/2, and powers like (x+y)^3.
Quick check
Expand (a - b)^2.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Multiply, and check by letting each letter equal 1 (or 2).
Multiply x + 4 by x + 3.
Multiply 2x + 3 by x + 3.
Multiply 4x + 7 by 3x + 2.
Multiply 3x - 5 by 3x + 8.
Multiply 3x - 2 by 2x + 3.
Multiply 6 - 4a by 5a - 7.
Multiply 2x + y by x + 3y.
Multiply 2x - 3y by 3x - 2y.
Multiply 3x - \tfrac{1}{2} by 2x - \tfrac{1}{3}.
Multiply -3x + 11a by 5x - a.
Multiply ax - bx by cx + dx.
Multiply -cx + d by bx - cx^2.
Multiply 4x - \tfrac{1}{3} by 6x + \tfrac{2}{3}.
Multiply x^2 - 5x + 6 by x - 3.
Multiply 3x^2 - 3x - 7 by 2x + 4.
Multiply x^2 - xy + y^2 by x + y.
Multiply a^2x^2 - 2a^2x + 4a^2 by ax + 2a.
Multiply 3x^3 - x^2 - 5x by 2x^3 - 5x^2.
Multiply 2x^2 - 7x + 12 by x^2 - 3x - 5.
Multiply a^2 - \tfrac{1}{2}a + \tfrac{1}{4} by a^2 - a + \tfrac{1}{3}.
Multiply x^2 - xy + y^2 by x^2 + xy + y^2.
Multiply 3x^3 + 5x^2 - x + 2 by x^2 - 2x + 1.
Expand (multiply out and combine like terms).
Expand (x^3 - x - 5)(2x^2 - 3x - 4).
Expand (3x - x^3 + x^2 - 6)(5 - x^2 - 3x).
Expand (4a - 5a^2 + 7 + a^3)(6 + a^2 - a + a^2).
Expand (5x - 4 + 8x^3)(8 - 5x^2 + 2x^3 - 9x).
Expand (x^2y - y^2x)(4xy - 5x^2y)(3x^2y - 7xy^2).
Expand (x^2 + y^2 + z^2 - xy - xz - yz)(x + y + z).
Expand (a + b + c)^2.
Expand (c + d - \tfrac{1}{2})^2.
Expand (a - 2b + 3c - 4d)^2.
Expand (x + y + z)^3.
Expand (x + y)^3 + (x - y)^3.
Expand (2a - b + 3)^2.
Expand (2x - 3ay)^3.
Expand (x + 2y)^2 - (x - 2y)^2.
Expand (4x - 3y)^2 - (3x + 4y)^2.
Expand (x - 3)^3 - (2x - 1)^2.
Expand (x^{a} - 3)(x^{a} + 4).
Expand (x^{2a} + 5)^2.
Expand (2x^{a} - 3)^3.
Expand (2x^{2a} - 3x)^2.